Their lightning start proved to be the ideal platform for a first win in seven outings, but they were almost caught out by a Norwich side who had previously scored just one goal on their travels.
The Clarets were ahead within a minute, when from City's own corner Kyle Lafferty broke before feeding Wade Elliott on the right, where his cross was headed in by an unmarked Robbie Blake.
Three minutes later the home fans were in dreamland when Andy Gray was fouled in the box by Jon Otsemobor. The former Sunderland man picked himself up to confidently dispatch the penalty.
Lafferty then shot straight at David Marshall, while at the other end David Strihavka sent two close-range headers over.
Luke Chadwick almost pulled one back straight before the break when his driven shot was saved instinctively by Gabor Kiraly.
Burnley had more than enough chances at the start of the second half to put the game out of sight as Blake fired a left foot shot wide and Lafferty forced a good low stop by Marshall.
But just after the hour, the Canaries grabbed themselves a lifeline when substitute John Hartson played a neat ball into the path of Chris Brown, who beat Kiraly with the aid of the post.
Hartson then had a header saved by Kiraly, before the Hungarian produced another good low save to deny the West Brom loan man.
Lafferty should have sealed the points with a free header that was put wide, but Burnley had already done enough to seal a much needed three points.