The visitors kicked off and David Marshall was soon in action saving a low shot from Neal Bishop.
However, the home side went ahead after just two minutes when Jamie Cureton converted Jon Otsemobor's right-wing cross with a crisp right-foot shot from 12 yards which beat Lee Harrison's dive to his right.
Cureton struck again in the 16th minute. Lee Croft's low cross from the right was blocked on the six-yard line but fell kindly for Cureton off the heels of Chris Brown and he drove the ball past the helpless Harrison.
Chris Brown's shot hit the top of the bar and bounced over before City went 3-0 up in the 21st minute through Simon Lappin who beat Harrison high to his left with a left-foot shot from 20 yards.
Mark Fotheringham then latched on to a loose ball and hit a low shot from 20 yards for Norwich's fourth goal and it was 5-0 after half an hour when Harrison blocked Cureton's shot but Darel Russell 's right-foot shot from seven yards gave him no chance.
In the second-half Brown made way for David Strihavka and Ian Hendon was replaced by Max Porter.
To their credit the Bees kept buzzing and on 65 minutes Jason Puncheon beat Marshall low to his left with a firm left-foot half volley from 16 yards.
The Canaries were stung again in the 74th minute when Adam Birchall's low shot from 18 yards beat Marshall to his right but the night ended in disappointment for Barnet when Bishop was sent off five minutes from time for a foul on Fotheringham.