The Seasiders looked set to add another memorable chapter to their cup folklore - almost 60 years after Sir Stanley Matthews inspired their sole triumph - when Ludovic Sylvestre gave them a shock lead on the hour mark.
But veteran Greece midfielder Karagounis sent a stunning half-volley past Matt Gilks with ten minutes remaining to earn a replay for the Cottagers.
Blackpool's Thomas Ince started despite reportedly being on the brink of re-joining Liverpool, as did captain Alex Baptiste - said to be close to a move to Leicester.
Baptiste was in the thick of it early on, beating Dimitar Berbatov to Karagounis' fine ball only to almost turn it past his own goalkeeper. And it was not long before Ince got in on the act, blasting too close to David Stockdale from close range after a neat 13th-minute move and lay-off from Chris Basham.
Basham then lived up to his name by clattering into Bryan Ruiz, bringing a booking and almost further punishment when the victim's 20-yard free-kick was tipped wide by Gilks.
Back came the visitors, Sylvestre thwarted by Steve Sidwell's desperate tackle before a lovely backheel from the recalled Kevin Phillips set up Ince for another drive Stockdale could only palm behind.
Fulham's frustration was growing, their attempts becoming increasingly speculative before half-time, with manager Martin Jol waiting just nine minutes into the second half before throwing on Hugo Rodallega for Ashkan Dejagah.
The home side had certainly upped the tempo but their dominance was an illusion as Blackpool broke out to take the lead on the hour mark. Nathan Delfouneso had just seen a penalty appeal turned down when he again found himself on the left edge of the area, this time picking out Sylvestre, who rifled a low 18-yard drive into the corner of the net.
Cue a Fulham siege, Blackpool defending desperately to deny the likes of Ruiz and Berbatov before Jol sent on Kieran Richardson for Alex Kacaniklic and Kerim Frei for Matthew Briggs.
With ten minutes remaining, Blackpool thought they had cleared a corner but they did not reckon with Karagounis, who controlled the ball before unleashing an unstoppable dipping half-volley from the corner of the area, which sailed over the helpless Gilks.
It could easily have been followed by a winner but Ruiz's header from Richardson's cross hit the bar and then there was an injury-time miss - seemingly by Rodallega - from a corner as the Seasiders held on.