Neither team had much to play for heading into the climax of the campaign after Fulham's relegation from the Premier League was confirmed with a 4-1 loss to Stoke City last weekend.
But they provided terrific entertainment in a frenetic contest in the last top-flight game at Craven Cottage for at least a season.
Dwight Gayle - the hero of Palace's remarkable 3-3 draw with Liverpool on Monday that had huge ramifications in the Premier League title race - looked to have given his side the points with a double, which was separated by Cauley Woodrow's first goal for Fulham.
However, substitute David came off the bench to ensure a share of the spoils late into added time with a magnificent curling effort that flew into the top corner.
Palace finish the campaign in 11th place, while Fulham, by contrast, end a depressing year second bottom.
Fulham created the first real chance after seven minutes, Alexander Kacaniklic's low drive drawing a near-post save from Palace goalkeeper Wayne Hennessey.
Palace crafted an opening themselves four minutes later, as Gayle played in Yannick Bolasie, only for Fulham shot-stopper David Stockdale to produce a fine save.
However, Stockdale was beaten just a few moments later when Gayle flicked home from Joel Ward's low cross into the area.
The visitors continued to dictate matters, and would have doubled their advantage had Stockdale not pulled off another strong stop to keep out Scott Dann's header from Tom Ince's free-kick.
Fulham did threaten an equaliser in first-half injury time, but Fernando Amorebieta's header was kept out by Hennessey at his right-hand post.
Amorebieta was heavily involved again as Fulham went agonisingly close to levelling early in the second half, the Venezuela defender's header back across goal nodded against the post by Hugo Rodallega.
Palace were forced into a substitution in the 59th minute as Mile Jedinak was replaced by Kagisho Dikgacoi after suffering an injury.
And Jedinak's absence was felt two minutes later as Woodrow netted the equaliser with an excellent curling finish beyond Hennessey after collecting a clever reverse pass from Rodallega.
Tony Pulis' men responded well, though, and appeared to have claimed the points late on as Gayle struck with a terrific low free-kick into the bottom corner from some way out.
Gayle then missed a glaring chance to claim his hat-trick, as he diverted the ball wide of an open goal after Glenn Murray had crashed an effort against the crossbar.
And that miss proved costly as David - who had come on for the injured Kacaniklic in the 23rd minute - ensured that Fulham ended a disastrous year on a high with a magnificent maiden goal for the club.
