© PA Photos
After the interval both sides went into attack mode but couldn't carve out any goalscoring chances until the latter stages.
As the match entered the final 15 of the 90 both keepers produced saves to keep the match deadlocked. First Gilks again kept out Rodallega before Mark Schwarzer tipped Thomas Ince's volley away.
The hosts took the lead in the 82nd minute when Nathan Delfouneso stroked into an empty net and that appeared to be the winner.
However, with the last kick of the game Kieran Richardson curled home a 25-yard effort to send the match into extra time.
Blackpool began the extra period on top and Schwarzer had to be alert to deny Gary Taylor-Fletcher before Ince smashed against the upright.
Both sides went for the winner to avoid a penalty shootout and it was Fulham who claimed their place in round four when Hangeland rose above the Tangerines' defence to get the slightest touch on Karagounis's corner.
Martin Jol's men will now face either West Ham or Manchester United in the next stage.