Chelsea keeper Petr Cech guessed correctly to parry Dempsey's 93rd-minute spot-kick after another game to forget for new £50million Blues striker Fernando Torres.
Torres missed four chances either side of half-time as champions Chelsea failed to climb back into the Premier League top four and they are now 12 points behind leaders Manchester United.
Manager Ancelotti axed Didier Drogba after he and Torres failed to ignite in the previous weekend's surprise defeat to Liverpool.
They were almost caught out again in the opening six minutes by a low Damien Duff cross but David Luiz - making his full debut - read the danger to beat Andy Johnson to the ball.
John Terry was thwarted by good pressure attempting to head Branislav Ivanovic's cross goalward, while there was also more than one blocked effort from the visitors.
Torres, though, was once more struggling to make an impact, with some of his many mistakes greeted by chants of 'What a waste of money' from the Fulham fans.
The in-form Duff surprised Cech in the 24th minute with a 30-yard effort the goalkeeper could only punch clear - the first save of any note.
Frank Lampard should have at least forced a second four minutes later when Chelsea's best move of the first half saw him sweep Ivanovic's low cross over the bar.
The latter was providing the only real width for the champions, whose next cross almost caught out a back-pedalling Mark Schwarzer moments after Florent Malouda's rising 25-yard attempt sailed just over.
Terry produced a vital glancing header at the other end before Ivanovic became the first player booked for tugging Moussa Dembele's shirt on the edge of - and arguably inside - the box.
Steve Sidwell soon followed for an ugly thigh-high lunge on Michael Essien.
With seconds of the half remaining, Torres saw two great chances to open his Chelsea account go begging, the first with the weakest of finishes after a strong run.
The second was gilt-edged, with the Spaniard's attempt to gather Luiz's glorious long ball simply horrible.
Ancelotti resisted the urge to turn to Drogba during the interval but Torres soon gave him cause to rue that decision by heading Ramires' pinpoint cross off target.
Chelsea wasted another opportunity when they took an indirect free-kick too quickly after Schwarzer picked up a Danny Murphy backpass.
And Torres' misery deepened when he curled high and wide after being released by a fine Lampard pass.
Supportive chants from the visiting fans rained down on the misfiring Spaniard, but to no avail.
Torres was completely outshone by fellow new boy Luiz, with the 23-year-old by far the best player on the pitch as Chelsea began to dominate without finding the breakthrough.
Ancelotti made a change with 25 minutes remaining, rubbing salt into Drogba's wounds by sending on Salomon Kalou for Nicolas Anelka.
Malouda failed to capitalise on an inch-perfect Ashley Cole ball as Fulham held firm going into the final 20 minutes.
Luiz should have capped his performance with a goal when the Brazilian rose completely unmarked to nod Malouda's corner over the top moments before Drogba finally replaced Torres.
Fulham then became even more compact, bringing on Simon Davies for Johnson as Chelsea began to be restricted again to long-range efforts.
But Essien should have given them a 77th-minute lead when he mistimed his diving header from Lampard's dangerous cross and Drogba showed what Chelsea had been missing with a barnstorming run down the left and a ball his team-mates should have anticipated.
Dempsey almost made Chelsea pay when he pounced on a Terry error only to see his shot beaten away by Cech.
Fulham ran down the clock by introducing Eidur Gudjohnsen and Zoltan Gera for Murphy and Dembele before an jaw-dropping overhead-kick cross from Luiz was headed over by Kalou.
The changes revitalised the hosts and Davies glanced a cross from Gera - who was booked in stoppage time - over the top.
In the third and final minute of added time, Luiz was adjudged to have tripped Dempsey in the box.
The American blasted the resulting penalty goalward but Cech saved and Dempsey scissor-kicked the rebound over the top.