Leeds had looked set to give their hopes of going up a further shot in the arm when Tim Ream's early own goal saw them leading going into the final stages.
However, the visitors were reduced to 10 men in the 89th minute when Kalvin Phillips was dismissed for a second caution, and Fulham took advantage as Cairney curled home a long-range strike to salvage a draw for the hosts.
The point apiece will do little for either side's promotion hopes, though. Fourth-placed Leeds still had an eye on reaching the top two while Fulham, in seventh, needed to make up ground to get in the play-off berths.
But Fulham were good value for at least a draw on a night when they may feel they should have walked away with all three points.
Garry Monk's visitors were second best from the moment Fulham defender Ream somehow miscued into his own net after five minutes.
It was some blunder by the American defender to jolt Fulham, who could ill-afford to drop points as they played catch-up on their play-off rivals.
The free-scoring hosts finally mustered an effort in the 23rd minute when Cairney fired inches wide after Neeskens Kebano's pass.
Fulham came even closer 30 seconds later when Kebano's run and powerful shot bounced down and out off the underside of the crossbar.
Fulham appealed to referee Lee Probert that the ball had gone in but, with no goal line technology in the Championship, play was waved on. Replays later suggested the ball had just crossed the line.
It was all Fulham as Scott Malone's long-range shot was well gathered by Leeds goalkeeper Rob Green in the latest bombardment of the visiting goal.
Leeds had created little but a quick counter-attack saw Alfonso Pedraza foiled by David Button.
Fulham began the second half as they finished the first, on top with the lively Kebano hitting the side-netting.
But Fulham were always culpable of a defensive mistake and this time it was an error from Tomas Kalas that nearly saw Kemar Roofe capitalise.
Button came to his side's rescue with a routine save but Fulham were finding it increasingly difficult to find a way back amid resolute defending from the Yorkshire outfit.
Leeds could have killed off the game in the 74th minute when the ball fell kindly to Pedraza, but he saw his shot hit the outside of the post.
Cairney saw his free-kick turned aside by Green in the 80th minute and Scott Parker blazed over, but Fulham looked to be running out of ideas.
That was until Phillips was handed his marching orders in the 89th minute for a second yellow card, and Cairney would have the final say when he fired home deep into added time.
Additional reporting by the Press Association.