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.
The draw means both sides will go into the hat for the fifth round on Monday, ahead of a replay at Craven Cottage.
Poyet made three changes from the side beaten at Tottenham last weekend as Vito Mannone was given a rare start in goal, with Defoe handed a home debut alongside Steven Fletcher in attack.
The visitors opted to leave midweek hat-trick hero McCormack on the bench and Fulham began to threaten after soaking up early Sunderland pressure.
Woodrow was first to test Mannone with an effort from the edge of the penalty area after 14 minutes, with Seko Fofana heavily involved in their attacking play.
The young Frenchman was at the heart of their next foray forward midway through the half as he managed to find Woodrow again, but Sunderland scrambled his effort to safety.
As the hosts continued to struggle for momentum, Dembele brought another save out of Mannone, his low shot easily held by the Italian goalkeeper.
Sunderland had failed to create any clear chances as half-time approached before Sebastian Larsson finally forced Bettinelli to make a sharp save with his legs from a low drive.
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.
