The three-goal margin flattered the Clarets ultimately, but Steve Cotterill's side were full value for the win.
There was an early scare for Burnley when Dion Dublin hit the bar from Robert Eagle's free-kick but Burnley bossed the first half with Wade Elliott, James O'Connor and Eric Djemba-Djemba off target with sighters from outside the box.
With Djemba-Djemba dominating midfield, the Clarets got their noses in front on the half-hour when the Cameroon international fed Michael Duff and, from his cross, Ade Akinbiyi thumped a header inside the near post.
Norwich came more into the game after the break, although Andy Gray felt he should have had a penalty when he was taken down by Jason Shackell eight minutes after the restart.
Chris Martin curled a shot just wide at the other end before Brian Jensen made a terrific save to deny Dublin from a Lee Croft corner.
Burnley doubled their lead with four minutes to play when O'Connor's shot was tipped onto the post by Tony Warner, only for Gray to bundle in his 11th goal of the season and first since November 28.
Burnley put the icing on the cake two minutes from time when Chris McCann ducked under Jon Harley's cross and Elliott arrowed home a fierce drive from the edge of the box for his second goal in three games.