The home side had chances to add to Jamie Cureton's goal after 20 minutes before Steven Thompson equalised nine minutes before half-time.
The Canaries dominated the second half but failed to conjure up a winning goal.
Norwich kicked off attacking the Barclay End goal but made little headway in the first 15 minutes.
Clarets keeper Brian Jensen then had to be quick off his line to deny Wes Hoolahan a shooting chance.
However, City's man of the match played an important part in his team's goal. His through ball found Mark Fotheringham and his low cross from the right was fired low into the net by Cureton, beating Jensen low to his left.
The Burnley keeper then pulled off a good save to keep out Carl Cort's downward header and did even better to block a fierce angled drive from Jon Otsemobor, when the Norwich right-back burst into the penalty area from the right wing.
The Clarets drew level in the 36th minute when David Marshall blocked Robbie Blake's shot, but Thompson followed up and found the net from close range with a crisp finish.
Five minutes into the second half Hoolahan tried his luck from long range but his shot cleared the bar.
The home side then took the game to Burnley, without getting in a really telling cross from either flank.
In the 63rd minute Fotheringham did bring a save from Jensen with a low shot and ten minutes later Gary Doherty headed Sammy Clingan's free-kick to Jason Shackell, but his weak shot was easily saved by Jensen.
In the last minute of the game Burnley substitute Chris Eagles fired over with only Marshall to beat, but had he scored it would have been hard luck on struggling City.