Gus Poyet's men have only won twice at the Stadium of Light this season in all competitions and that run was extended as the visitors earned a fourth-round replay.
Fulham were much the better side in the first half and would have been ahead had Cauley Woodrow and Moussa Dembele taken good openings.
With home chances at a premium, Sunderland fans had failed to see anything from new signing Jermain Defoe until he was denied on three occasions by Marcus Bettinelli around the hour mark.
Jack Rodwell's red card after 65 minutes handed Fulham the initiative, but Ross McCormack and Hugo Rodallega were unable to find a way through.
Any momentum they may have created before the break disappeared early into the second half as Fofana got back on the ball, the 19-year-old seeing a low shot deflected narrowly wide.
Defoe's first notable involvement came after 55 minutes as he won the ball in midfield before driving towards goal, although his effort was sliced wide and never going to threaten Bettinelli.
The former Tottenham striker thought he had opened his Sunderland account six minutes later as he connected with a half-volley from close range, only for Bettinelli to push it onto the post.
Rodwell was then given his marching orders as he blocked Bettinelli's attempt to set Fulham on a counterattack to earn a second yellow card, his first coming after a high tackle in the opening 45 minutes.
McCormack's introduction gave Kit Symons' side a new dimension going forward, but it was Rodallega who went closest to winning the match as his late header thudded against the post, leaving Sunderland grateful to earn a replay.
