However, it was Darren Huckerby who created most of the chances and would have got on the scoresheet himself but for the heroics of Brian Jensen in the Burnley goal.
The visitors kicked off with Canaries' fans expecting great things from loan signings Crouch, Huckerby and Kevin Harper.
Crouch threatened twice in the first 15 minutes with a low shot saved by Jensen and a header that cleared the cross bar.
His best chance to impress caem in the 18th minute when he headed wide from Harper's right-wing free-kick.
Five minutes later Harper volleyed inches wide from Gary Holt's left-wing cross.
Robbie Blake fired inches wide at the other end before City forced four corners in two minutes on the half hour without seriously troubling Jensen.
Huckerby had two chances to open the scoring immediately after the break. First he fired straight at Jensen, before dribbling round the keeper only to run the ball out of play.
Damien Francis brought a fine save from Jensen with a low snap shot, and when Huckerby tried to put the midfielder in for another chance he over-hit his pass.
The Clarets had Roche was booked for fouling Huckerby and then, on exactly 58 minutes, Crouch calmly beat Jensen to his left with a low right-foot shot from 12 yards after the Burnley defence failed to clear Adam Drury's cross from the left.
With 20 minutes to go Crouch and Harper made way for Alex Notman and Iwan Roberts.
In the closing stages Francis fired in a stinging shot which Jensen beat away and the keeper twice denied Huckerby with similar saves.
And in the final minute the lively Huckerby set up Roberts with a low left-wing cross to enable the veteran striker to seal victory with a right-foot shot inside Jensen's left hand post.