Nathan Delfouneso's goal eight minutes from full time looked to have given Blackpool a narrow win on a freezing night at Bloomfield Road.
But substitute Kieran Richardson drilled an unstoppable shot past Matt Gilks deep into stoppage time to hand Martin Jol's side a lifeline.
Both sides had chances to settle the tie in extra time until Brede Hangeland headed in a Damien Duff corner with just four minutes remaining.
It was harsh on the Seasiders and their caretaker boss Steve Thompson, who may not be the favourite to land the job on a full-time basis but certainly coaxed a performance from his side.
Fulham, who left out Dimitar Berbatov and Bryan Ruiz, created their first serious chance after a nice passing move in the 14th minute but Mladen Petric sliced well wide from distance.
Blackpool were finding possession harder to come by but might have done better on the counter-attack when Gary Taylor-Fletcher curled a shot into Mark Schwarzer's arms. Chris Baird was inches away from giving the visitors the lead 10 minutes before the break when he met a Giorgos Karagounis corner with a free header but glanced it across goal.
The bulk of the second half was light on quality and goalmouth action. Hugo Rodallega almost broke the deadlock when a Mladen Petric flick found him in front of goal but Gilks palmed his shot wide.
That stirred Blackpool into life for the remaining 15 minutes and Chris Basham had a low shot from a Thomas Ince cross deflected narrowly wide of the post. The Tangerines did not dwell on that misfortune and pushed forward again until a Stephen Crainey cross led to confusion in the Fulham area and Schwarzer ran into bodies as he tried to punch clear. The goalkeeper made only poor contact on the ball and Delfouneso was on hand to tap the ball home from close range.
Yet just as it seemed Fulham's night was over, a miss from Blackpool's Tiago Gomes proved costly as late pressure from Jol's men paid off. The game was deep into the third minute of injury time when the ball was pushed out to Richardson outside the box and he unleashed a firm shot which left Gilks standing.
Unperturbed by the late blow, Blackpool made a positive start to extra time with Ince twice testing Schwarzer with long-range drives. Schwarzer tipped the first of them away for a corner and the second was seized upon by Taylor-Fletcher, but Delfouneso could not make the most of his cross.
Richardson threatened to repeat his earlier heroics for Fulham only to shoot wide but Fulham's opportunity came after Karagounis won a corner. Duff swung in the cross from the right and defender Hangeland headed the ball past Gilks for the winner, with the possible aid of a deflection.
Fulham will travel to the winners of Wednesday's replay between Manchester United and West Ham.