The Cottagers stunned Roberto Mancini's men with a second-half comeback.
City looked set to equal their best-ever opening to a top-flight campaign when Sergio Aguero's jaw-dropping adaptation to life in England continued with the seventh and eighth goals of his five-game City league career.
He made it 2-0 just seconds after the break today but Fulham refused to buckle, with Bobby Zamora and Danny Murphy earning them a deserved draw.
The result was the first blow to City's title challenge and, coming four days after their first ever Champions League game, it cast doubt on the ability of Mancini's men to combine the rigours of domestic football with that of Europe's top club competition.
Sunday's game was the fourth time in the last four rounds of Premier League fixtures that City had kicked off before Manchester United.
They had been trumped by their arch-rivals on each occasion but arrived in west London determined to lay down another marker on a ground where they won 4-1 last season.
Fulham went like for like, manager Martin Jol keeping faith with his 4-2-3-1 system as he sought his first Premier League win in charge.
City went close to opening them up inside seven minutes after Edin Dzeko strayed offside when put through by Samir Nasri.
They continued to boss proceedings, and although Clint Dempsey sent a 25-yard drive straight down the throat of Joe Hart after 15 minutes, City duly took the lead three minutes later.
Gareth Barry found David Silva on the edge of the box, and he was allowed to turn and feed Aguero, who timed his run perfectly before sidefooting home.
Mark Schwarzer denied City a second goal three times in the space of three minutes, beating Dzeko to Aguero's cross, tipping Clichy's 20-yard shot wide and getting fingers to Dzeko's piledriver after another great move involving Yaya Toure and Silva.
The recalled Zamora ballooned a 30-yard shot into the stands as Fulham continued to be given the runaround, while Dempsey looked their only real threat, drilling another effort too close to Hart.
The frustrated home fans roared on encouragement as their side enjoyed a rare spell of possession but Fulham posed little threat to the City goal.
Jol emerged for the second half with a face like thunder, but whatever he said during the break appeared to be rendered void when Aguero doubled the visitors' lead straight from the kick-off.
Clichy pumped a diagonal ball forward to Dzeko, whose knockdown was fired into the bottom left corner by the Argentinian.
Aguero thought he had completed back-to-back Premier League hat-tricks - he hit one against Wigan eight days ago - three minutes later with an identical finish but he was narrowly offside.
The home side took full advantage of their reprieve when they finally created a gilt-edged chance in the 55th minute.
The build-up was of the City kind, ending when Dempsey threaded in the unmarked Zamora, who curled the ball into the top corner.
Zamora had been played onside by Micah Richards, who redeemed himself moments later when his last-ditch challenge stopped Dempsey making it 2-2.
Fulham's first-half lethargy had vanished and Barry was booked for chopping down Mousa Dembele, whose low drive was palmed wide by Hart after Pajtim Kasami replaced an ailing Damien Duff.
Hart excelled himself again by tipping over Dempsey's bullet header from a corner before Mancini took action, withdrawing Silva for Pablo Zabaleta midway through the half.
The visitors remained a threat on the break, Richards glancing wide from a corner.
But it was Fulham's pressure that told when they equalised 15 minutes from time, Danny Murphy's scuffed 20-yarder taking a huge deflection off Vincent Kompany and going beyond the stranded Hart.
Craven Cottage was rocking and Mancini was forced to make a change again when Adam Johnson and Carlos Tevez replaced Nasri and Aguero in quick succession.
The fresh legs almost told in the final five minutes when Tevez was denied by a brave Brede Hangeland block.
Sidwell was booked and Jol threw on Bryan Ruiz for the excellent Dembele but Toure almost had the final say when his 30-yard strike drifted wide.