Norwich will be wondering why they didn't have the game sewn up in the first quarter, as the visitors were made to pay for a series of missed chances.
First of the chances arrived as early as the first minute when Jon Harley headed off the line from Leon McKenzie after the Norwich striker robbed defender Michael Duff of possession.
Then it was Darren Huckerby's turn to miss a straightforward opportunity as he got goal side of full-back Phil Bardsley, but was denied by a fine block from Brian Jensen.
The first corner of the game saw Norwich create another opportunity as Jonatan Johansson's header from a Simon Charlton cross beat Jensen, but struck the crossbar.
Sixty seconds later Nigel Worthington's team wasted a glorious opportunity as Frank Sinclair's header fell to McKenzie clean through on goal, but his tame lob was easily snaffled up by a grateful Jensen.
Completely against the run of play, the home side scored with their first attempt on goal, as a low cross from Graham Branch was controlled by Andy Gray.
With his back to goal, Gray spun and shot in one movement past the helpless Robert Green.
Gray was making his Turf Moor debut following his signing from Sunderland and his goal brought to an end Burnley's goal famine as the Clarets had not scored in their previous five games.
Another Turf Moor debutant, right-back Bardsley, went close to extending the lead with a fine attempt from 20 yards which flashed wide.
Bardsley was a key figure in Burnley's second goal six minutes into the second half as he collected a pass from Michael Ricketts and crossed for Branch to head down into the net and secure the points for Steve Cotterill's side.