Chris Hughton's side went into the tie having not won any of their last 11 games against Fulham, and were beaten 2-1 at home by Rene Meulensteen's men just 11 days ago in the Premier League.
A similar outcome looked to be on the cards when Darren Bent opened the scoring for the visitors after 40 minutes.
However, Fulham could not hold on to their lead and Robert Snodgrass deservedly equalised for the hosts just before the interval.
Norwich then generally controlled the play after the break, with Josh Murphy and Ricky van Wolfswinkel spurning inviting opportunities, but Fulham managed to earn a replay.
Hughton made eight alterations to the Norwich team that drew at Crystal Palace in their last Premier League match, with fit-again striker Van Wolfswinkel making his first start since October 6.
Meulensteen made nine changes of his own following Fulham's win over West Ham, including the re-introduction of Clint Dempsey after the USA international returned to the club on loan from Seattle Sounders.
Both sides could only threaten from range in the early stages, as Norwich's Murphy curled just over from the edge of the area on his full debut before Dempsey narrowly missed the target from 30 yards.
Teenager Murphy went close again eight minutes before the break when his curling effort was well saved by David Stockdale, but Fulham took the lead shortly after as Bent headed Chris David's cross past the helpless Mark Bunn.
Stockdale then denied Van Wolfswinkel as Norwich went in search of an immediate equaliser, but the hosts did have to wait long to level as Murphy terrorised Elsad Zverotic down the left before seeing his cross converted by Snodgrass' diving header on the stroke of half-time.
The second half started as the first ended; with Murphy wreaking havoc on the left flank. The 18-year-old winger got in behind Zverotic again before whipping the ball across, but he just missed the goal and the lurking Van Wolfswinkel.
Norwich were gradually starting to control proceedings and, with just under an hour gone, Stockdale was forced into another good stop as he palmed away Van Wolfswinkel's excellent header.
The hosts looked intent on avoiding a replay, but Fulham's willingness to hold out for a draw was obvious as they invited pressure on to themselves in the final 15 minutes.
Hughton's side pressed for a late winner, but ultimately the visitors held firm to secure a rematch at Craven Cottage.
‘Arteta, Alonso, Emery, me… none of us were physical players – we needed the understanding of the game. That probably helped us move into management’: Premier League boss reveals reasons for natural career progression
‘England have the players to win the World Cup – it’ll be tough for Thomas Tuchel to do a bad job, with the squad he has at his disposal’ Former Three Lions winger backs new boss after gentle qualifying draw
