The Canaries kept their play off hopes alive for another week thanks to goals from Steen Nedergaard and Zema Abbey.
Robbie Blake did his best to get something from the game for the Lancashire side but his colleagues failed to match his enthusiasm.
The home side kicked off attacking the River End goal and went ahead in the third minute when Nedergaard got his head to Paul McVeigh's left-wing corner and the ball beat Nik Michopoulos at the near post.
Abbey missed the chance to increase the lead when his shot was blocked by a defender with the keeper out of position.
The visitors seemed to take heart from this escape and put Norwich under some pressure, Blake bringing a good save from Robert Green after 15 minutes.
The Burnley keeper did equally well to keep out David Healy's free-kick a minute later.
Mark Rivers rescued City with a goal line clearance from a cross deflected goalwards by one of his defenders in the 20th minute.
Lee Briscoe was carried off with a leg injury to be replaced by Gordon Armstrong after 26 minutes.
Darren Kenton was booked for a late tackle on Paul Weller. Gareth Taylor volleyed over the bar and Blake made an impressive dive but failed to earn a penalty.
Branch was booked for tripping Healy two minutes before the interval and Blake's free-kick was well saved by Green in stoppage time.
Healy appeared to have been tripped by the Burnley keeper but referee Tomlin annoyed the home fans when he waved play on.
In the 55th minute substitute Armstrong was substituted by Richard Chaplow.
Abbey calmed the Norwich nerves when he scored a second goal for the home side in the 67th minute, stabbing the ball home from close range after Michopoulos could only block a first shot from the lively Healy.