Manager Antonio Conte had made nine changes from the 2-2 draw with Arsenal in the league on Wednesday, keeping only Gary Cahill and Tiemoue Bakayoko in the team.
Norwich, relegated to the Championship (second tier) two years ago after one season back in the Premier League, had slightly the better of a quiet first half.
Goalkeeper Angus Gunn saved well from Willian in one of the few clear-cut chances either side made, and bringing on Alvaro Morata from a youthful substitutes' bench did not have the desired effect for Conte's side.
They now face a replay in London to add to a busy January, which also includes a two-leg League Cup semi-final against Arsenal.
(Reporting by Steve Tongue; Editing by Christian Radnedge)
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