The Londoners led 2-0 after 35 minutes through Dimitar Berbatov's audacious penalty and Sascha Riether's close-range finish.
However, Craig Gardner, who had conceded the spot-kick, dragged the Black Cats back into it when he too converted a penalty before the break, and Sessegnon prevented his side from slipping to a fourth successive Barclays Premier League defeat when he fired home with 20 minutes remaining.
Both sides could have won it in a frenetic conclusion with Adam Johnson furious not to be awarded a third penalty of the afternoon when his shot appeared to strike substitute Emmanuel Frimpong's arm, but they had to make do with a point apiece.
Martin Jol, whose Fulham side arrived on Wearside three points better off than their hosts, could be forgiven for thinking they should have been leaving with a victory, but the Black Cats were just about value for their draw after a woeful stat to the game.
Sunderland desperately needed a win after taking just one point from their last four games, but for more than half an hour, they looked painfully incapable of achieving it.
They found themselves in deep trouble by the time 35 minutes had elapsed and worryingly for manager Martin O'Neill, Fulham barely had to break sweat to claim their advantage.
They took the lead with 16 minutes gone when Gardner, to his astonishment and that of the locals, was adjudged to have tripped Ashkan Dejagah inside the box with referee Mark Halsey taking an eternity to point to the spot.
Berbatov's execution from 12 yards was little short of outrageous as he stuttered up to the ball before side-footing it to the left of keeper Simon Mignolet, who was rooted to the spot as he attempted to second-guess the Bulgarian.
It could have been worse for the Black Cats nine minutes later when Dejagah got in on the left, only to fire across goal, but the respite was temporary.
Johnson's 35th-minute corner was cleared by, of all people, Berbatov and as the home side slept, Fulham broke at pace and in numbers before Mignolet could only parry Dejagah's shot from a tight angle to Riether, who gleefully tapped home.
Sunderland, who had gone close through John O'Shea's second-minute header, were in complete disarray, but were handed a way back into the game within two minutes.
Central defender Philippe Senderos literally got to grips with striker Danny Graham as the pair challenged for a ball over the top and Mr Halsey awarded a second penalty to equal surprise.
Where Berbatov had opted for guile, Gardner chose power to smash a rising shot - his side's first on target - past Mark Schwarzer to give them hope.
The home side were instantly rejuvenated as they set about the task of restoring parity, and Sessegnon might have levelled with a minute of the first half remaining when he unleashed a right-foot shot which Schwarzer bundled around the post.
O'Neill's men headed for the dressing room at least having taken the edge off the manager's half-time team-talk, but knowing they needed to improve drastically if they were to turn the game around.
They might have been back on level terms within two minutes of the restart when Steven Fletcher and Graham combined for the fist time of note in the game, but the former Swansea frontman scuffed his shot tamely at Schwarzer.
Berbatov blasted high and wide from distance after finding space on the right with 49 minutes gone, but the visitors continued to pose the greater threat with Berbatov in imperious form.
Alfred N'Diaye went close twice within a minute, firing just too high from distance and then heading well over Schwarzer's crossbar, but the Black Cats were starting to cause problems on a sustained basis.
The game would have effectively been over had Mignolet not raced from his line to block Berbatov's effort after Bryan Ruiz had played him in with a lofted pass, and the save proved crucial within seconds when Sessegnon drilled home from 15 yards after the ball broke invitingly for him.
Fletcher was inches away from getting a touch to Gardner's driven 75th-minute cross, but Berbatov passed up a glorious opportunity to win it seconds later when he shot weakly at Mignolet.
Johnson was convinced Frimpong had handled his 79th-minute drive and it took a point-blank block by Brede Hangeland to deny Fletcher a headed winner, but Sunderland were at last able to end their damaging losing run.