The managerless Championship side looked to have dumped out their Premier League visitors when Nathan Delfouneso struck after 82 minutes on a freezing cold night at Bloomfield Road.
But Kieran Richardson came off the bench to rescue Fulham in injury time and Hangeland headed home in the 116th minute to book a date with Manchester United or West Ham.
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.
The hosts had shown confidence from the start and Fulham goalkeeper Mark Schwarzer had an uneasy moment when he punched away a Tom Ince cross unconvincingly.
Nathan Eccleston tried to curl the ball back past the Australian but Philippe Senderos got his head in the way.
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.
The Seasiders threatened again when Gary Taylor-Fletcher launched a long ball forward but Ince was beaten to the ball by Schwarzer.
Fulham were angry not to be awarded a corner when Kirk Broadfoot clearly headed an Ashkan Dejagah cross behind his own goal, but referee Andre Marriner awarded only a goal kick.
Dejagah had a chance himself after exchanging passes with Sascha Riether in the box but failed to extend Matthew Gilks.
Hugo Rodallega did better when the ball fell to him from a Petric corner but Gilks saved his low shot with his feet. The Colombian headed over from the resulting corner.
Blackpool were finding possession harder to come by but might have done better on the counter-attack when Taylor-Fletcher curled a shot into 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 hosts pushed as the first half came to a close but Broadfoot headed over from a corner and Delfouneso undid a good move by blasting well over.
Ince had the first opening after the break but shot tamely at Schwarzer from the edge of the area after breaking through.
The bulk of the second half was light on quality and goalmouth action.
Dejagah provided some of the more illuminating moments and released Riether on the right with a superb backheel on the hour but Petric could not make the most of his cross.
Dejagah tried to force the issue again when he powered forward soon after but shot wide of target.
Rodallega almost broke the deadlock when a 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 an Ince cross deflected narrowly wide of the post. For a second time in the game, Marriner mistakenly awarded a goalkick.
The Tangerines did not dwell on that misfortune and pushed forward again.
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.
Schwarzer complained to no avail and the visiting defence was almost caught out again when substitute Tiago Gomes beat Baird but shot into the side-netting.
Yet just as it seemed Fulham's night was over, the Gomes miss proved costly as late pressure from Martin 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.
A glorious chance opened up after Gomes powered forward again to test Schwarzer with a long-range drive, but Broadfoot, having charged from deep, blazed the rebound over. Dejagah shot at Gilks as Fulham responded and Rodallega headed over.
Blackpool had two more chances early in the second period but Schwarzer stood up to deny Ince from a long Ludovic Sylvestre pass and then dealt with a Taylor-Fletcher header.
Richardson threatened to repeat his earlier heroics for Fulham only to shoot wide but Fulham's opportunity came after Karagounis won a corner.
Damien 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.
- Sports & Recreation
- Mark Schwarzer
- Matthew Gilks