It ended the club's six-match unbeaten run in European competition, cost them top spot in the group and meant they squandered the chance to qualify for the knockout phase with two games to spare.
Mancini's men are now level with Lech on seven points but drop into second place in the table having scored less goals than the Poles.
Juventus's draw with Salzburg in the Group A's other match means City require only three points from their remaining two fixtures to guarantee a place in the last 32 of the competition.
However, that is unlikely to silence the critics Mancini has accused of writing "lies" about his team, who travel to West Brom on Sunday and host Manchester United three days later.
The Italian made eight changes to the side that started at Wolves in the Premier League on Saturday but City were up against a team they beat 3-1 at Eastlands two weeks earlier and whose preparation was disrupted by the sacking of coach Jacek Zielinski on Wednesday.
Zielinski's replacement Jose Maria Bakero was part of the Barcelona "Dream Team" that won the 1992 European Cup but he had little time to prepare his players and they were completely outplayed in the opening quarter.
Seemingly unflustered by the deafening noise that greeted them at the Stadion Miejski, City pushed forward in numbers from the outset and saw several clear cut chances come and go.
A dazzling run from Adam Johnson ended with James Milner being foiled by left-back Luis Henriquez and goalkeeper Jasmin Buric palmed Adebayor's near-post shot to safety after Micah Richards and Shaun Wright-Phillips combined well on the right.
City were set up in a 4-2-3-1 formation but full-backs Richards and Wayne Bridge attempted to join the attack whenever possible and Lech found themselves struggling to cope.
In the 18th minute City bombarded their goal, Buric repelling a powerful Adebayor header and Patrick Vieira twice going close before Dedryck Boyata met Johnson's centre with a weak header.
That barrage appeared to galvanise the hosts and suddenly City were on the back foot, Slawomir Peszko drilling a low free-kick inches wide and Semir Stilic lashing off target moments later.
Lech were stirring and, with City struggling to keep hold of possession, the pressure soon told.
The visitors could not clear Sergei Krivets's left-wing cross and when Vieira's header found its way to Injac, the midfielder rifled an unstoppable drive into City goalkeeper Shay Given's bottom right-hand corner.
Mancini cut a beleaguered figure as his side slumped to defeat in Poland
The visitors showed little desire to redress the balance before half-time - their body language particularly poor - but Mancini replaced the ineffective Wright-Phillips with Silva at the interval and they looked an altogether different prospect.
They started passing and moving with a real sense of purpose and when Adebayor slammed the ball home after his header from Milner's corner was parried, the scores were level.
But back came Lech and Rudnevs saw a shot saved, Peszko had a strong penalty appeal turned down and Jacek Kielb watched a drive fly narrowly wide.
At the other end Silva's volley bounced into the ground and on to the bar after a fine run and cross by Adebayor.
Substitutes came and went as both managers seemed content with a draw, but the Polish champions then benefited from a piece of fortune to retake the lead.
Boyata met Krivets's free-kick from the left with a clearing header but it ricocheted off the head of Arboleda and into the net.
And City's misery was complete when substitute Mozdzen let fly with an spectacular strike that sent the home crowd wild with joy.
90:00+4:10 The game is over, as the referee blows his whistle.
90:00+3:50 The ball is sent over by Pablo Zabaleta, clearance by Manuel Arboleda.
90:00+3:40 Unfair challenge on Aleksandar Kolarov by Jakub Wilk results in a free kick. Direct free kick taken by Aleksandar Kolarov.
90:00+2:03 Foul by Adam Johnson on Jakub Wilk, free kick awarded. Jasmin Buric takes the free kick.
90:00+0:28 The assist for the goal came from Siarhey Krivets.
90:00+0:28 GOAL - Mateusz Mozdzen:Lech Poznan 3 - 1 Man City Mateusz Mozdzen fires in a brilliant goal from outside the penalty area to the top left corner of the goal. Lech Poznan 3-1
89:41 Adam Johnson produces a cross, David Silva takes a shot from close range clearing the crossbar.
88:08 Pablo Zabaleta has an effort at goal from outside the area missing to the wide right of the target. Corner from the right
by-line taken by David Silva, clearance by Marcin Kikut. Long range effort by Aleksandar Kolarov goes wide right of the target.
87:39 Foul by Jakub Wilk on Adam Johnson, free kick awarded. Aleksandar Kolarov crosses the ball in from the free kick, clearance
by Bartosz Bosacki.
85:30 Dedryk Boyata gives away a free kick for an unfair challenge on Artjoms Rudnevs. The free kick is swung in right-footed by
Siarhey Krivets, clearance by Dedryk Boyata.
85:30 GOAL - Manuel Arboleda:Lech Poznan 2 - 1 Man City Manuel Arboleda finds the back of the net with a headed goal from inside the penalty area. Lech Poznan 2-1 Man City.
84:26 The assistant referee flags for offside against Emmanuel Adebayor. Indirect free kick taken by Jasmin Buric.
83:58 Artjoms Rudnevs fouled by Joleon Lescott, the ref awards a free kick. Free kick taken by Marcin Kikut.
82:38 A cross is delivered by Adam Johnson, Foul by David Silva on Luis Henriquez, free kick awarded. Direct free kick taken by
81:48 Jacek Kielb concedes a free kick for a foul on Micah Richards. Direct free kick taken by Dedryk Boyata.
80:14 David Silva gives away a free kick for an unfair challenge on Jakub Wilk. Direct free kick taken by Marcin Kikut.
79:16 Ivan Djurdjevic gives away a free kick for an unfair challenge on Micah Richards. Free kick taken by David Silva.
79:04 Manuel Arboleda challenges Emmanuel Adebayor unfairly and gives away a free kick. Direct free kick taken by David Silva.
77:56 Mateusz Mozdzen has an effort at goal from just outside the box which goes wide of the left-hand post.
77:14 Substitution Vincent Kompany joins the action as a substitute, replacing James Milner.
76:29 Inswinging corner taken by Jakub Wilk from the right by-line, Joleon Lescott makes a clearance.
76:22 Luis Henriquez sends in a cross, Aleksandar Kolarov makes a clearance.
75:50 A cross is delivered by Jacek Kielb, Dedryk Boyata makes a clearance.
74:51 Pablo Zabaleta concedes a free kick for a foul on Ivan Djurdjevic. Ivan Djurdjevic takes the free kick.
72:02 Substitution Slawomir Peszko goes off and Jakub Wilk comes on.
71:32 David Silva takes a shot.
71:11 Shot by Siarhey Krivets. Blocked by Joleon Lescott.
70:30 David Silva delivers the ball.
70:09 Bartosz Bosacki challenges James Milner unfairly and gives away a free kick. Pablo Zabaleta takes the direct free kick.
69:14 Foul by Pablo Zabaleta on Siarhey Krivets, free kick awarded. Marcin Kikut restarts play with the free kick.
69:14 Substitution Aleksandar Kolarov comes on in place of Wayne Bridge.
67:09 Jacek Kielb produces a right-footed shot from just outside the area that goes wide left of the goal.
63:31 Micah Richards crosses the ball, Bartosz Bosacki manages to make a clearance.
63:06 The referee blows for offside. Direct free kick taken by Joleon Lescott.
61:44 Free kick awarded for a foul by Bartosz Bosacki on Emmanuel Adebayor. James Milner takes the free kick.
61:44 Substitution Mateusz Mozdzen is brought on as a substitute for Semir Stilic.
61:13 Siarhey Krivets delivers the ball, clearance made by Dedryk Boyata.
61:00 Shot from 12 yards by Artjoms Rudnevs. Save by Shay Given.
60:09 Free kick awarded for an unfair challenge on Jacek Kielb by Patrick Vieira. Shot comes in from Slawomir Peszko from the free
59:29 Unfair challenge on Marcin Kikut by James Milner results in a free kick. Jasmin Buric takes the direct free kick.
58:07 Pablo Zabaleta gives away a free kick for an unfair challenge on Slawomir Peszko. Siarhey Krivets takes the direct free kick.
57:26 James Milner crosses the ball, Emmanuel Adebayor takes a shot. Jasmin Buric makes a comfortable save.
54:39 Ivan Djurdjevic gives away a free kick for an unfair challenge on Pablo Zabaleta. Adam Johnson restarts play with the free
54:39 Booking Ivan Djurdjevic booked for unsporting behaviour.
53:28 Foul by Artjoms Rudnevs on Wayne Bridge, free kick awarded. Shay Given takes the free kick.
52:24 Free kick awarded for a foul by Pablo Zabaleta on Artjoms Rudnevs. The free kick is swung in left-footed by Semir Stilic,
save by Shay Given.
51:22 Substitution Dimitrije Injac goes off and Jacek Kielb comes on.
50:27 Corner taken right-footed by James Milner, Emmanuel Adebayor takes a shot. Save by Jasmin Buric.
50:27 GOAL - Emmanuel Adebayor:Lech Poznan 1 - 1 Man City Emmanuel Adebayor fires in a goal from inside the six-yard box to the top left corner of the goal. Lech Poznan 1-1 Man City.
49:27 Outswinging corner taken from the right by-line by James Milner.
48:28 Semir Stilic fouled by Pablo Zabaleta, the ref awards a free kick. Free kick crossed left-footed by Semir Stilic.
47:40 Shot by Semir Stilic. Joleon Lescott gets a block in.
47:14 James Milner delivers the ball, save made by Jasmin Buric.
46:32 Foul by Emmanuel Adebayor on Dimitrije Injac, free kick awarded. Free kick taken by Ivan Djurdjevic.
45:23 Pablo Zabaleta challenges Slawomir Peszko unfairly and gives away a free kick. Free kick taken by Marcin Kikut.
45:01 The game restarts for the second half.
45:01 Substitution Shaun Wright-Phillips leaves the field to be replaced by David Silva.
45:00+0:35 It is the end of the first-half.
45:00+0:30 Booking The referee cautions Artjoms Rudnevs for dissent.
44:58 Corner taken by Semir Stilic played to the near post, clearance made by Wayne Bridge.
43:16 Foul by Luis Henriquez on James Milner, free kick awarded. The free kick is delivered left-footed by Adam Johnson from right
40:57 The referee blows for offside. Pablo Zabaleta takes the indirect free kick.
39:01 Wayne Bridge gives away a free kick for an unfair challenge on Slawomir Peszko. The free kick is delivered left-footed by
Semir Stilic from right wing.
39:01 Booking The referee shows Wayne Bridge a yellow card for unsporting behaviour.
37:12 Micah Richards concedes a free kick for a foul on Siarhey Krivets. Free kick taken by Ivan Djurdjevic.
37:12 Booking Micah Richards is booked.
33:06 Inswinging corner taken by Siarhey Krivets from the left by-line.
29:53 GOAL - Dimitrije Injac:Lech Poznan 1 - 0 Man City Dimitrije Injac finds the net with a goal from outside the box to the bottom left corner of the goal. Lech Poznan 1-0 Man
29:18 Wayne Bridge delivers the ball, clearance by Manuel Arboleda.
28:27 Unfair challenge on Ivan Djurdjevic by Patrick Vieira results in a free kick. Jasmin Buric restarts play with the free kick.
27:41 Patrick Vieira fouled by Ivan Djurdjevic, the ref awards a free kick. Direct free kick taken by Patrick Vieira.
26:36 Micah Richards gives away a free kick for an unfair challenge on Luis Henriquez. Free kick taken by Jasmin Buric.
25:50 Adam Johnson fouled by Luis Henriquez, the ref awards a free kick. Direct free kick taken by Dedryk Boyata.
22:55 Effort from outside the penalty box by Semir Stilic goes wide right of the goal.
21:45 Foul by James Milner on Slawomir Peszko, free kick awarded. Shot comes in from Slawomir Peszko from the free kick.
19:41 Free kick awarded for a foul by Artjoms Rudnevs on Dedryk Boyata. Dedryk Boyata takes the direct free kick.
17:49 Dedryk Boyata takes a shot. Save by Jasmin Buric.
Correction - 17:33 Emmanuel Adebayor takes a shot. Dimitrije Injac manages to make a clearance.
17:33 Emmanuel Adebayor takes a shot. Marcin Kikut manages to make a clearance.
17:09 Outswinging corner taken left-footed by Adam Johnson from the left by-line.
16:21 Wayne Bridge crosses the ball, comfortable save by Jasmin Buric.
15:13 Outswinging corner taken by James Milner, save by Jasmin Buric.
14:48 Outswinging corner taken right-footed by James Milner played to the near post, clearance made by Artjoms Rudnevs.
14:13 Free kick awarded for a foul by Dimitrije Injac on James Milner. Micah Richards takes the direct free kick.
13:53 Effort from 35 yards by Luis Henriquez missed to the right of the net.
13:46 The ball is sent over by Marcin Kikut, Micah Richards manages to make a clearance.
12:49 Inswinging corner taken left-footed by Semir Stilic from the right by-line to the near post.
8:55 The ball is sent over by Adam Johnson, Marcin Kikut gets a block in.
6:43 Emmanuel Adebayor takes a shot. Save made by Jasmin Buric. Corner taken by James Milner from the right by-line, clearance
by Manuel Arboleda. The assistant referee flags for offside against Joleon Lescott. Jasmin Buric restarts play with the free
5:26 The official flags Artjoms Rudnevs offside. Free kick taken by Joleon Lescott.
3:07 James Milner takes a shot. Manuel Arboleda gets a block in. Foul by Semir Stilic on Dedryk Boyata, free kick awarded. Wayne
Bridge takes the direct free kick.
0:46 The referee blows for offside. Direct free kick taken by Dedryk Boyata.
0:00 The game begins.
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