Collins John over-shadowed his new strike partners Andy Cole and Tomasz Radzinski to earn Fulham a point.
The 18-year-old Dutchman turned a sharp finish past David James after 57 minutes to level after Robbie Fowler gave City a first-half advantage.
Fowler hooked home a spectacular finish after 28 minutes to hand City a deserved lead before Collins struck.
City were then denied by a string of saves from Edwin van Der Sar, who twice denied Antoine Sibierski spectacularly.
Fulham improved after a dismal first-half show, but Kevin Keegan's City will still feel they did enough to earn all three points on the opening day.
Shaun Wright-Phillips, who has agreed a new four-year contract, lit up the start to the season at the City of Manchester Stadium.
He showed a blistering turn of speed to leave the Fulham defence trailing, and it needed a desperate clearance to prevent Nicolas Anelka taking advantage.
It set the tone for a bright City display in the opening 45 minutes and they took a deserved lead through Fowler after 28 minutes.
Danny Mills' long throw was headed on by Sibierski and Fowler lifted a left-foot finish high past Van der Sar.
Fulham had their new strike force of Cole and Radzinski on show, but they barely had a kick as City dominated.
But in a rare attack, Fulham snatched an undeserved equaliser after 55 minutes.
Claus Jensen's cross found John unmarked, and the young Dutch striker had time and space to turn and score past the flat-footed David James.
City were suddenly on the back foot, but Van der Sar was forced to turn over a rising drive from Claudio Reyna, then clutch a Dunne header.
Sibierski should have put City back in front after 67 minutes, but his point-blank header was blocked by Van der Sar.
Fulham posed a threat on the break, and it needed a fine save from James to deny Jensen with ten minutes left.
Man City: James, Mills, Dunne, Distin, Thatcher,
Wright-Phillips, Reyna, Bosvelt (Barton 78), Sibierski,
Fowler (Macken 77), Anelka.
Subs Not Used: Stuhr-Ellegaard, Jihai, McManaman.
Goals: Fowler 28.
Fulham: Van der Sar, Volz, Pearce, Knight, Bocanegra, Diop,
John (McBride 63), Legwinski, Radzinski (Pembridge 79), Cole,
Subs Not Used: Crossley, Rehman, Goma.
Booked: Cole, Volz.
Goals: John 56.
Ref: M Messias (S Yorkshire).