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Olivier Giroud's double - his first home Premier League goals - could not prevent Arsenal staying seventh in the table, sliding four points behind fourth-placed Everton who came from behind to beat visiting Sunderland 2-1.
"I'm proud of the way we played because we started not so well. It would have been very harsh if they had scored the penalty," Fulham manager Martin Jol told Sky Sports after Mikel Arteta wasted the chance to hand Arsenal a streaky win.
Surprise packages West Bromwich Albion, in fifth, are level on points with Everton following a 2-1 victory at Wigan Athletic thanks to James Morrison's header and a Gary Caldwell own goal.
Leaders Manchester United can stretch their one-point advantage when they visit Aston Villa later (1730 GMT) given second-placed Chelsea and Manchester City, in third, host Liverpool and Tottenham Hotspur respectively on Sunday.
Former Liverpool midfielder Charlie Adam's first goal for Stoke City sealed a 1-0 home win over Queens Park Rangers who plunged to the bottom of the table after Southampton leapfrogged the Londoners with a 1-1 draw against Swansea City.
Reading and Norwich City drew 0-0 in a dour game.
Arsenal have stuttered all season and lost at Manchester United in the league last weekend after an insipid performance followed by a Champions League draw at Schalke 04 which complicated their qualification hopes.
Arsene Wenger's men got off to the perfect start against Fulham when close-season signing Giroud finally began to show some of the might which helped him win the French title last term with Montpellier.
The tall striker netted with a powerful header on 11 minutes after a corner from the recalled Theo Walcott and the hosts were soon 2-0 up and cruising when Arteta set up Germany's Lukas Podolski for an easy finish midway through the first half.
Fulham were down to 10 men at the time after an injury to Kieran Richardson.
The unmarked Berbatov, a nemesis of Arsenal as a Manchester United player, pulled one back on the half hour when he headed in a corner as the home team again flattered to deceive.
The laidback Bulgarian then set up substitute Alex Kacaniklic in his own inimitable style as Fulham equalised five minutes before the break.
Berbatov put Fulham 3-2 in front to the gasps of the Arsenal crowd, who had booed their team off at the break, when he converted the coolest of penalties after Bryan Ruiz was pulled back by Arteta in the 67th minute.
Giroud then took another small step in his quest to replace former Arsenal talisman and now United hero Robin van Persie by producing another good header from a Walcott cross two minutes later after the Frenchman had first hit the post.
Arsenal were awarded a stoppage-time penalty for a harsh-looking handball but Mark Schwarzer saved Arteta's spot kick to seal a 3-3 draw, the latest high-scoring game to grace the upper echelons of English football.
Fulham stayed eighth and level on points with Arsenal whose fans are getting increasingly restless and putting Wenger under pressure like never before.
Everton have been in great form but looked to be heading for defeat against Sunderland when Adam Johnson netted on the stroke of halftime to become the first player other than Steven Fletcher to score for the visitors in the league this term.
No one has epitomised Everton's drive this season more than Belgium midfielder Marouane Fellaini and he fittingly grabbed the equaliser by drilling the ball home on 76 minutes.
Three minutes later Fellaini played in Nikica Jelavic for the winner.
The victory marked David Moyes's 400th Premier League match as a manager as he joined a select group containing Alex Ferguson, Wenger and Harry Redknapp to reach the milestone.