Two goals from Danny Crow and a goal apiece from Josh Clarke and Olly Willis was more than enough to give Adrian Boothroyd's side the win, indeed the score line would have been much more in the Canaries favour had it not been for some great saves from Bristol's James Stabler.
Lee Blackburn, who was on target for the Reserves at Northampton on Wednesday, was the first to test Stabler while both Clarke and Crow went close in the opening minutes.
At the other end Joe Lewis brought off an instinctive save to deny Joe Anyinsah.
Greg Crane gets in an important challenge
David Chick covers a Bristol attack
The breakthrough came on 20 minutes when Crow sent a high ball to the far post where Clarke rose above Stabler to head the ball home.
The visitors went close to drawing level when Anyinsah saw a shot glance the upright but most of the action was in the Bristol half where late in the first-half Stabler denied Robert Eagle, Willis and Clarke.
City were still in dressing room when Bristol drew level just 20 seconds into the second half, Callum Hart crossing for Joe Long to finish at the near post.
City restored their lead just 5 minutes later when Crow drove a left foot shot into the far corner.
Crow then had a goal disallowed and Stabler made yet another save from the City front man, but Crow would not be denied his second and City's fourth after some good work by Damian Batt who crossed for Crow to finish from just inside the area.
Lee Blackburn in the thick of things
City Line-up: Lewis, Batt, Chick [J.Osborne 80], Crane [Watts 80], Tyrie, Smith, Willis [Howell 80], Blackburn, Clarke, Crow, Eagle.
Subs not used: Howlett, Herbert.
Top Canary: Danny Crow.
