Sunderland claimed a home win at the final attempt to avoid being the first team in English league history to end a season without a home win.
Le Tallec's header was his fifth goal of the season
Anthony Le Tallec's glancing header put them ahead in the first half.
Justin Hoyte missed a great chance for 2-0 when he trod on the ball but Chris Brown made no mistake when he poked home a fumbled Mark Crossley save.
Tomasz Radzinski's curling shot gave Fulham hope but Sunderland held on in a tense finish for a much-celebrated win.
Fulham were leading through a Brian McBride goal when the original fixture in April was abandoned in a blizzard, and they came into the match seeking a fourth straight win, which they have never previously managed in the Premiership.
But their chances took a knock early on when Luis Boa Morte went off with what looked like a thigh injury, having fallen heavily after a foul by Andy Welsh.
There was little goalmouth action in the first half until Sunderland went ahead, with Welsh a key player in the move.
He linked up well on the left with George McCartney, who fired in a cross, glanced into the far corner by Le Tallec's diving near-post header.
A rush of blood from Kelvin Davis nearly allowed Radzinski in to level for Fulham, but fortunately for the keeper the attack was snuffed out and turned into a counter-attack.
Nyron Nosworthy raced up the right wing and crossed to Hoyte, but he summed up Sunderland's season when he trod on the ball and fell over as the ball dribbled safely behind him.
Nosworthy was a key player in the second goal when it came, as Fulham keeper Crossley could only block his shot straight into the path of the advancing Brown, who netted.
The novelty of being in control of the match seemed to lessen Sunderland's concentration and it nearly proved costly when Fulham almost scored twice in as many minutes.
Radzinski curled in a shot from the edge of the box to pull one goal back before Collins John came within inches of an equaliser when he barged Gary Breen out of the way but put his shot just the wrong side of the post.
The tension rose as the clock ticked down, but Sunderland managed to hold on for a deserved and longed-for victory.
Sunderland caretaker boss Kevin Ball:
"From a selfish point of view, it's nice but what's most important is our fans, players and everyone connected with the club.
"Beforehand, I said 'Let's send people home with smiles on their faces'. I'm pleased to be able to report they did.
"The supporters were magnificent and got behind us when we needed them to."
Fulham manager Chris Coleman:
"For 65 minutes we didn't play very well. In the last 20 minutes when we were 2-0 down, we made chances, but it was too little, too late.
"I'm disappointed at the fashion in which we lost the game but I can't be too hard on the guys.
"We were playing a very determined team. They were ready for it and better than us, it's as simple as that."
Sunderland: Davis, Nosworthy, Breen, Caldwell, McCartney, Hoyte (Kyle 74), Whitehead, Miller, Welsh, Brown, Le Tallec (Daryl Murphy 81). Subs Not Used: Alnwick, Danny Collins, Leadbitter.
Booked: Caldwell, McCartney, Brown, Whitehead.
Goals: Le Tallec 32, Brown 57.
Fulham: Crossley, Rosenior, Christanval, Pearce (Brown 69), Bridge, Malbranque, Diop, Elliott, Boa Morte (Radzinski 18), Helguson, McBride (John 60). Subs Not Used: Warner, Volz.
Booked: Elliott, Christanval, Pearce, Bridge.
Goals: Radzinski 76.
Ref: M Riley (West Yorkshire)