Both sides struggled to find an opening in the first 20 minutes, with the first chance coming just before the half-hour mark.
The Cottagers had another chance shortly afterwards when youngster Kerim Frei found space on the edge of the box before firing a low shot that forced Cech into a smart save.
Right on the stroke of half time Chelsea made a rare foray into the Fulham penalty box, where Salomon Kalou was brought down by Danny Murphy. Frank Lampard stepped up and scored his 150th Premier League goal, a record for a midfielder.
The Blues began the second half as they had finished the first, with Lampard going close from 25 yards out before having another penalty claim turned down.
The hosts sides began to ease themselves back into the game and almost equalised when John Arne Riise was released down the left and tried an audacious effort with the outside of his boot that drifted just wide.
Fulham got their equaliser with less than ten minutes left, and it came courtesy of Dempsey once again as he rose highest to nod in Riise's corner, via a deflection off Gary Cahill, moments after Cech had denied Aaron Hughes's goalbound header.
There was still time for Dempsey slide Riise through on the goal, but the former Liverpool man's touch was too heavy the chance went begging.
The result means that Chelsea now sit sixth in the Premier League, two points behind Spurs and Newcastle, who are fourth and fifth respectively, while Fulham move up two spots to ninth.