Fulham were on the wrong end of a controversial penalty decision when Salomon Kalou went down under Stephen Kelly's challenge, with Frank Lampard converting from the spot on the stroke of half time.
Petr Cech pulled off a stunning save to deny Aaron Hughes but Fulham were rewarded for their pressure moments later when Dempsey leapt high to head in, although the ball took a big deflection off Gary Cahill.
The result leaves Chelsea in sixth place, two points behind Tottenham and Newcastle.
Following Tottenham's shock defeat to Norwich and Newcastle's earlier victory over Bolton, there was pressure on Chelsea to stay in contention for a Champions League place with victory in West London.
But Martin Jol’s side created the best chances early on as Fulham threatened to sneak ahead at a windswept Craven Cottage.
Moussa Dembele has been a revelation since joining from AZ Alkmaar and his trickery got him past Branislav Ivanovic, but Dempsey could only stab the Belgian’s pull back straight at Cech.
Fulham continued to attack and the impressive Kerim Frei almost capped a rare start with a memorable goal, weaving into space 25 yards out and unleashing a venomous low drive that forced Cech to turn the ball around his post.
But an absorbing first half swung Chelsea’s way when Kalou went down in the box after Kelly and Danny Murphy combined to bring him down.
It appeared soft but Lampard dispatched the penalty into the bottom corner to notch up his 150th Premier League goal – the first midfielder to reach that milestone.
Chelsea, who left Juan Mata on the bench, came flying out of the blocks at the start of the second half and had a legitimate penalty claim turned down when Lampard appeared to be impeded by Brede Hangeland as the England midfielder made a darting run to reach Ryan Bertrand’s low cross.
But Fulham weathered the storm and enjoyed their brightest moment when Frei, Fulham’s best player on the night, slid a perfectly-weighted through ball into John Arne Riise, whose audacious effort with the outside of his left foot drifted just wide of the far post.
Fulham had the wind in their sails at this point and came agonisingly close to levelling when Hughes’s thumping header was somehow clawed out by Cech, whose phenomenal save underlined his world-class credentials once again.
But there was nothing Cech could do when Dempsey finally found the net moments later. Rising high, the unmarked Texan glanced in Damien Duff’s corner.
There was a certain inevitability about Fulham’s equaliser, but they could have snatched an unlikely win late on when Riise was played in by Dempsey, but a heavy touch by the Norwegian allowed Cahill to recover.
Chelsea now find themselves two points off Spurs and Newcastle - and this will go down as a missed opportunity in the Blues' pursuit of Champions League football.