For injury-hit Fulham it was a welcome - and ultimately comfortable - win but their two decisive first-half goals could both have been ruled out.
The Cottagers had scored only once in their last five league games but could be encouraged by the visitors' extraordinarily inept away form.
The Clarets arrived in west London with just a solitary point from their previous 12 away matches, while conceding at a rate of about three goals a game.
The clearest opening in the scrappy early exchanges fell to Damien Duff in farcical scenes that would prove ominous for the visitors.
The Irish midfielder found himself in acres of space down the right-hand side but looked almost surprised to be deemed onside - and duly wasted the chance to cross.
Hodgson delighted with comfortable win
The Clarets looked to be matching their lacklustre opponents until another poor decision from the referee's assistant led to Brian Laws' side falling behind.
Nicky Shorey's lofted pass from the left found Elm, who looked a clear foot ahead of the last defender before he headed the ball into the path of the onrushing Murphy.
The Fulham captain deftly controlled with his chest before superbly firing a volley past Jensen.
Steven Fletcher almost levelled instantly when his sizzling left-footed effort flew just past the post, before the Cottagers scored another disputed goal.
This time, Zamora was the man who appeared offside when he raced on to Chris Baird's pass down the line.
Zamora was allowed to cut inside and his shot was parried by Jensen straight to the feet of Elm, who calmly dispatched the ball past the helpless Clarets keeper.
Burnley boss Laws immediately introduced winger Chris Eagles for injured defender Leon Cort as the Clarets attempted to hit back.
The closest they came was from free-kicks, with a 25-yard Tyrone Mears effort grazing the crossbar and Danny Fox hitting a set-piece straight at Mark Schwarzer.
Zamora almost added a third goal after the break when the ball fell kindly to him the area but Jensen got down superbly to palm the striker's low effort around the post.
Laws questions inconsistent decisions
However, it didn't take long for Fulham's top scorer to get on the scoresheet. After David Edgar felled Zamora on the edge of the area, the striker picked himself up to place a low free-kick past Jensen.
If the keeper felt disappointed not to get a hand on the shot, he partially redeemed himself minutes later by saving well with his legs from substitute Stefano Okaka.
Burnley battled to find a way back into the game and Eagles missed two good chances to add some tension to the closing stages - but Fulham's eighth home win of the season never looked in danger.
Burnley's away record will not look any healthier after another comprehensive defeat on the road, but Laws will be left wondering what might have been if the game's vital decisions had gone his way.
Fulham manager Roy Hodgson "We made the win comfortable with the quality of our play, particularly our defending.
"We were good value for our two goals at half-time and Burnley will be grateful their keeper was in good form.
"We've lost a lot of quality players through injury, all at the same time, so I'm pleased that February has started well."
Burnley manager Brian Laws "It's been clearly shown that the first two goals were offside, but we've contributed to that too with our defending.
"Even though we didn't play particularly well, you want consistency from the referee and the linesman.
"We know we have to change things away from home, whether it's the personnel or the formation."
90:00+3:09 The referee signals the end of the game.
90:00+0:33 Effort from deep inside the area by Stefano Okaka misses to the right of the target.
89:59 Outswinging corner taken by Bjorn Helge Riise, Clarke Carlisle manages to make a clearance.
89:47 Bjorn Helge Riise takes a short corner.
89:14 Bjorn Helge Riise takes a short corner.
88:39 Chris Eagles takes a shot. Save by Mark Schwarzer.
87:35 Daniel Fox takes a shot.
86:50 Unfair challenge on Nicky Shorey by Chris Eagles results in a free kick. Nicky Shorey takes the free kick.
86:04 Andre Bikey gives away a free kick for an unfair challenge on David Elm. Direct free kick taken by Danny Murphy.
83:27 The referee blows for offside against Stefano Okaka. Indirect free kick taken by Brian Jensen.
Correction - 80:37 Steven Fletcher takes a shot. Save by Mark Schwarzer.
80:37 Headed effort from inside the area by Steven Fletcher misses to the left of the target.
Correction - 80:11 Free kick awarded for an unfair challenge on Daniel Fox by Jonathan Greening. Direct free kick taken by Daniel Fox.
80:11 Foul by Jonathan Greening on Daniel Fox, free kick awarded. Daniel Fox restarts play with the free kick.
77:38 Stefano Okaka takes a shot. Save by Brian Jensen.
76:35 Substitution Bjorn Helge Riise joins the action as a substitute, replacing Damien Duff.
75:12 Substitution (Burnley) makes a substitution, with Martin Paterson coming on for David Nugent.
75:12 The offside flag is raised against David Elm. Brian Jensen takes the free kick.
71:56 Unfair challenge on David Nugent by Dickson Etuhu results in a free kick. Free kick taken by David Nugent.
70:36 Substitution Stefano Okaka is brought on as a substitute for Bobby Zamora.
67:11 Bobby Zamora is caught offside. Free kick taken by Brian Jensen.
66:45 Damien Duff takes a shot. David Edgar gets a block in.
66:25 Corner taken short by Jonathan Greening.
66:13 Damien Duff takes a shot. Kevin McDonald gets a block in.
64:07 Free kick awarded for a foul by David Elm on Jack Cork. Direct free kick taken by Brian Jensen.
62:12 Corner taken left-footed by Daniel Fox.
61:03 Free kick taken by Mark Schwarzer.
61:03 Substitution Jonathan Greening comes on in place of Simon Davies.
61:03 Andre Bikey gives away a free kick for an unfair challenge on David Elm.
58:03 Free kick awarded for a foul by Chris Baird on Steven Fletcher. Daniel Fox takes the direct free kick.
56:13 Chris Eagles takes a shot from deep inside the box clearing the bar.
55:23 Substitution Wade Elliott goes off and Jack Cork comes on.
54:49 Daniel Fox takes a outswinging corner.
53:33 GOAL - Bobby Zamora:Fulham 3 - 0 Burnley Free kick scored by Bobby Zamora from long range to the bottom left corner of the goal. Fulham 3-0 Burnley.
Correction - 53:21 GOAL - Bobby Zamora:Fulham 3 - 0 Burnley Bobby Zamora hits the back of the net with a goal direct from the free kick from outside the penalty box to the bottom left
corner of the goal. Fulham 3-0 Burnley.
52:23 Free kick awarded for an unfair challenge on Bobby Zamora by David Edgar.
Correction - 52:14 David Edgar gives away a free kick for an unfair challenge on Bobby Zamora.
50:02 Corner taken right-footed by Simon Davies.
49:50 Bobby Zamora takes a shot. Save by Brian Jensen.
49:09 Andre Bikey takes a shot. Save by Mark Schwarzer.
45:01 The second half has started.
45:00+2:09 The match has reached half-time.
45:00+1:35 Effort on goal by Simon Davies from just outside the area goes harmlessly over the target.
42:27 Bobby Zamora fouled by David Edgar, the ref awards a free kick. Direct free kick taken by Danny Murphy.
41:10 Free kick awarded for an unfair challenge on Steven Fletcher by Dickson Etuhu. Tyrone Mears shoots direct from the free kick.
39:47 Free kick awarded for an unfair challenge on Clarke Carlisle by David Elm. Direct free kick taken by Brian Jensen.
36:20 Direct strike on goal from the free kick comes in from Daniel Fox, save by Mark Schwarzer.
36:20 Booking Booking for Dickson Etuhu for unsporting behaviour.
36:15 Free kick awarded for an unfair challenge on Steven Fletcher by Dickson Etuhu.
35:32 Centre by Tyrone Mears, clearance made by Danny Murphy.
Correction - 33:58 Bobby Zamora is flagged offside by the assistant referee. Clarke Carlisle takes the indirect free kick.
33:58 The assistant referee signals for offside against Bobby Zamora. Direct free kick taken by Clarke Carlisle.
32:50 Unfair challenge on Simon Davies by David Edgar results in a free kick. Chris Baird takes the direct free kick.
31:54 Substitution Chris Eagles on for Leon Cort.
30:49 Bobby Zamora provided the assist for the goal.
30:49 GOAL - David Elm:Fulham 2 - 0 Burnley Goal scored by David Elm from close range low into the middle of the goal. Fulham 2-0 Burnley.
30:38 Bobby Zamora takes a shot. Save by Brian Jensen.
Correction - 28:34 Effort on goal by Dickson Etuhu from just outside the area goes harmlessly over the target.
28:34 Shot by Dickson Etuhu from just outside the area goes over the net.
28:22 Bobby Zamora takes a shot. Blocked by Leon Cort.
27:56 Dickson Etuhu takes a shot. Fantastic save by Brian Jensen.
27:05 Damien Duff fouled by David Edgar, the ref awards a free kick. Direct free kick taken by Aaron Hughes.
26:32 David Nugent is caught offside. Brede Hangeland takes the direct free kick.
24:36 Steven Fletcher produces a right-footed shot from just outside the penalty box that goes wide left of the target.
22:42 Assist on the goal came from David Elm.
22:42 GOAL - Danny Murphy:Fulham 1 - 0 Burnley Danny Murphy finds the back of the net with a goal from inside the penalty box to the bottom left corner of the goal. Fulham
22:08 Simon Davies decides to take a short corner.
21:24 Steven Fletcher has shot on goal from just outside the box which goes wide of the right-hand upright.
20:56 David Elm takes a shot. Save by Brian Jensen.
18:47 David Elm is ruled offside. Indirect free kick taken by Brian Jensen.
12:24 Andre Bikey has an effort at goal from just outside the area which goes wide of the left-hand post.
10:01 The official flags Bobby Zamora offside. Brian Jensen takes the free kick.
9:17 Danny Murphy takes a shot. Leon Cort gets a block in.
7:49 Damien Duff has an effort from just inside the box that misses to the left of the target.
7:27 Damien Duff fouled by David Edgar, the ref awards a free kick. Free kick taken by Danny Murphy.
5:20 The referee blows for offside. Indirect free kick taken by Mark Schwarzer.
4:07 David Elm gives away a free kick for an unfair challenge on Andre Bikey. Andre Bikey restarts play with the free kick.
2:10 Damien Duff fouled by Andre Bikey, the ref awards a free kick. Direct free kick taken by Danny Murphy.
1:20 Unfair challenge on Tyrone Mears by Simon Davies results in a free kick. Brian Jensen restarts play with the free kick.
0:00 The referee gets the game underway.
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