Both managers rung the changes as they remain focused on Premier League survival, with the Canaries' Scotland international canceling out Darren Bent's first-half opener.
Chris Hughton's side went into the tie having not won any of its last 11 games against Fulham, and was 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 visiting side after 40 minutes.
However, Fulham could not hold on to the lead and Robert Snodgrass deservedly equalized for the home team 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 its last Premier League match, with fit-again striker Van Wolfswinkel making his first start since Oct. 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 the 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 equalizer, but the home side did not have to wait long to level as Murphy terrorized Elsad Zverotic down the left before seeing his cross converted by Snodgrass' diving header on the stroke of halftime.
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 was 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 home side looked intent on avoiding a replay, but Fulham's willingness to hold out for a draw was obvious as the club invited pressure on to itself in the final 15 minutes.
Hughton's side pressed for a late winner, but ultimately the Cottagers held firm to secure a rematch at Craven Cottage.