Simeon Jackson earned Norwich a repeat of their Premier League result two weeks ago by scoring late on to beat West Brom in the FA Cup fourth round.
The visitors were thankful to debutant goalkeeper Jed Steer, who tipped Jerome Thomas's shot onto the bar.
Grant Holt then scored from close range following Elliott Bennett's cross.
Marc Antoine-Fortune equalised after Thomas dispossessed the Norwich defence but substitute Jackson came on to take advantage of Craig Dawson's error.
Dawson was dispossessed inside his own box and Jackson needed no second invitation to drill a low shot past Foster and into the far corner of the net.
Earlier, Fortune headed over at the near post from a corner and Simon Cox had a shot deflected over from Fortune's low centre.
But it was Steer, 19, who played a key role in Norwich's victory and prevented the tie going to a replay by saving late on from substitute Shane Long and Joe Mattock.
The goalkeeper's heroics booked Norwich's place in the fifth round for the first time in five years and extended their unbeaten run to six games.
The Canaries, who are ninth in the Premier League, will also take satisfaction that they progressed to the next round despite resting five players.
West Brom were hoping to reach the last 16 for the fourth time in six years, but must now concentrate on retaining their Premier League status.
Norwich manager Paul Lambert:
"I thought Jed Steer's saves were world class. He was terrific. It is only one game but he has a great temperament and a great enthusiasm for the game as well which is important.
"He is a really level-headed lad, which is probably more important than anything - but his handling was brilliant. His two saves at the end were exceptional."
West Brom manager Roy Hodgson:
"I am sad to lose because the performance was one where we did not deserve to lose the game, although we didn't really want to go to Norwich midweek for a replay.
"But now we can concentrate on our main priority, which is staying in the Premier League."