A gutsy second-half fightback saw Ne-Jai Tucker and his Burnley teammates score twice in eight minutes to just about keep their play-off hopes alive.
Two defensive errors handed Norwich a half-time lead and the prospect of a second successive defeat for Burnley.
But captain Lewis Richardson led the recovery when he forced the ball home at the second attempt with 15 minutes to go.
And substitute Dara Costelloe then poked home his first goal at Under-23 level as the Clarets added another point to their quest to secure a top-five finish.
With just two games to go it may not be enough to keep their promotion prospects in their own hands as they slipped a place to fourth in the Second Division of PL2.
Norwich hadn’t posed too many questions of their own but struck twice in five minutes to take the lead as Burnley were made to pay for taking their eye off the ball.
Central defender Richard Nartey has barely put a foot wrong all season in a stand-out campaign but was caught out by Tony Springett, who crossed for Abu Kamara to steer a curling effort into to the top corner to give the Canaries the lead.
And Nartey then failed to cut out a long clearance from goalkeeper Joe Rose – who had just made a vital save from Richardson following another run from the Burnley frontman – and Matt Dennis ran through to slot home City’s second.
Richardson replied with a stinging effort against the crossbar and then when Joel Mumbongo and Costelloe couldn’t take chances early in the second half, it looked as though the gap may be too big to bridge.
But Burnley’s, driven on from midfield by Anthony Gomez-Mancini, kept forcing Norwich back and got their reward with a quarter of an hour to go.
Richardson’s first touch from Tucker’s floated ball forward hit the post but the striker reacted well to nod home the loose ball.
Burnley goalkeeper Marc Richter was forced into his only real save of the afternoon to keep out an effort from City substitute Jon Rowe as Norwich looked to pounce on the break as Burnley committed men forward.
But Burnley’s pressure paid off for a second time when Costelloe levelled.
Again, Richardson was denied by the woodwork after being picked out by full-back Anthony Glennon – a constant supply line from the left flank.
But Burnley’s recycled the ball and when it dropped to Costelloe, the young Irishman kept his composure to prod the ball into the bottom corner.
Richardson then had another shot blocked as he tried to sniff out a winner, but it also needed a goalline clearance from Glennon to preserve a point in a dong-dong finish which may yet be pivotal in the final reckoning.