Martin Paterson marked his return to the side with a first-half brace - his first goals this term - to seemingly put Burnley on their way to three points and only their second victory in 27 league games on the road.
But substitute Chris Martin pulled one back for City after 71 minutes, netting his first goal in 10 games just eight minutes after replacing Simeon Jackson.
Anthony McNamee, who impressed after coming on at half-time, then drilled a shot against a post four minutes later before Crofts popped up to level from close range.
Crofts was then sent off moments later for his second booking, but his goal prevented Norwich from suffering back-to-back league defeats during a season for the first time under boss Paul Lambert.
Lambert kept faith with the same side having been impressed with the performance in the 3-1 defeat at Cardiff last Saturday.
Paterson replace Scotland international Chris Iwelumo as counterpart Brian Laws made one change to the side which earned a 1-1 draw at QPR.
Former Manchester United midfielder Chris Eagles, who has scored five goals in his last six games, had the first two opportunities of the game inside seven minutes.
First his 25-yard effort was comfortably saved by City goalkeeper John Ruddy, who then watched as Eagles' shot from the edge of the penalty area went past his post.
Leon Barnett headed a Simon Lappin corner over the bar before the Canaries had a great chance to take the lead in the 19th minute.
Burnley stopper Lee Grant denied Jackson and moments later made an even better save to keep out a header from Lappin, who should have done better.
The visitors then took a stranglehold on the game with two goals in the space of seven minutes.
A great ball by Wade Elliott found Tyrone Mears and his cross was tucked home by Paterson in the 26th minute.
Paterson then claimed his second as he beat Ruddy again after being played in by Dean Marney, the ball trickling over the line.
Lambert made a change at half-time with Lappin being replaced by McNamee, who was straight into the action and had an early shot cleared by the Burnley defence.
Norwich came close to cutting the deficit in the 50th minute but Grant produced another fine save to keep out Barnett's header.
The home side carved out another opportunity in the 68th minute and again it fell to centre-half Barnett, who headed Wes Hoolahan's cross over the bar.
But City did get back into the game with 19 minutes left when Martin produced an exquisite finish after a Crofts cut-back and McNamee was then foiled by the post.
The Canaries pressed for an equaliser and it came when Crofts pounced from six yards before he saw red.
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