Grant Holt and Simeon Jackson fired the Barclays Premier League side 2-0 ahead inside 12 minutes. However, Jay Rodriguez's looping header reduced the deficit - and then City skipper Holt missed a penalty.
The Canaries, though, were soon back in command when Andrew Surman swept home on the hour and substitute Steve Morison tapped in number four from close range.
Norwich manager Paul Lambert resisted the temptation to make wholesale changes to his staring XI, retaining the back four from the 2-1 win at Queens Park Rangers.
Reserve keeper Declan Rudd was given a chance to impress, and made two early saves as Clarets made a bright start with Ross Wallace and then Jay Rodriguez shooting from distance.
Norwich then took the lead after just six minutes.
Wes Hoolahan sent the ball across the Burnley penalty area from the left, and Holt arrived in the six-yard box to power a header past Lee Grant.
Despite the setback, Burnley - who lasted just one season in the top flight after winning promotion in 2009 - remained positive as a dangerous centre by Rodriguez was cleared by Daniel Ayala in front of his own net.
It was, though, soon 2-0 to the Canaries, who are currently perched ninth in the Premier League, following an impressing opening on their return to the top flight.
Again the problems were caused down the left, this time with veteran full-back Adam Drury getting clear on the overlap to cross for Jackson to nod in at the near post on 12 minutes.
However, Burnley were immediately back in the tie when Rodriguez got up to send a looping header into the far corner from Wallace's free-kick.
The tie was now shaping up to be an end-to-end contest rather than one-way traffic.
Norwich went back on the offensive, with Canada striker Jackson stretching the Burnley rearguard.
Norwich's pressure paid off when they were awarded a penalty in the 31st minute after Andrew Croft's cross struck Charlie Austin on the arm, and referee Mike Jones pointed to the spot following a signal from his nearside assistant.
Holt, who had never missed a penalty since joining Norwich in the summer of 2009, stepped up - only to drill his shot wide of the left-hand post.
At the other end, Rodriguez let fly from 25 yards, with Rudd down quickly to make the save.
The Norwich goalkeeper then had to be alert to rush out and clear an underhit backpass from Zak Whitbread, getting to the ball just ahead of Paterson.
Burnley were enjoying a decent spell and Ayala had to make a sliding tackle to block Austin in the six-yard box.
There was a stoppage on 50 minutes when Ben Mee and Wallace collided into each other after the ball bounced around the centre-circle.
Midfielder Wallace eventually had to be replaced by Marvin Bartley.
On 53 minutes, Republic of Ireland midfielder Hoolahan was denied by a fine one-handed save from Grant.
The ball dropped to Holt in the six-yard box, but his shot came back off the inside of the post, with Jackson then stabbing the rebound wide.
On the hour, it was 3-1.
Norwich broke quickly, and the referee allowed a good advantage after Jackson was bundled over on the edge of the penalty area which allowed Surman to charge through and lot past Grant.
Norwich made a double change with 19 minutes left as Jackson and Holt were replaced by Morison and Aaron Wilbraham.
The substitutes combined to make it 4-1 on 73 minutes, when Wilbraham's shot was saved by Grant, and Wales striker Morison tapped in his eighth goal of the season.
Burnley continued to look for a way back into the tie, but despite all of their endeavour it was Norwich who comfortably moved through to Sunday's fourth-round draw.