The Cumbrians went ahead in the 12th minute with a spectacular volley by Vincent Pericard.
Joe Anyinsah moved in from the right to provide the cross which looked to be going behind Pericard but the big striker surprised Norwich keeper Fraser Forster with an unorthodox shot from just inside the box for his fifth goal in 10 games.
However, once Norwich got their passing game going Carlisle surrendered the initiative and the visitors created openings down the left flank. It was from there that Wes Hoolahan cut in and supplied the cross for Grant Holt to beat keeper Adam Collin with a 26th-minute diving header at the near post - his 16th goal of the season.
Carlisle regained the lead in the first attack of the second half after Pericard beat Simon Lappin in the challenge and cut the ball back perfectly for Kevan Hurst to sweep a shot wide of Forster just inside the keeper's right-hand post.
The home side made it 3-1 in the 72nd minute. Graham Kavanagh's corner was blocked but the veteran midfielder returned the ball to the far post where Richard Keogh rose highest to nod home.