Dean Morgan struck two minutes after replacing the injured Carlos Edwards to seemingly put the Hatters on course for victory, after Rowan Vine had given them an early lead.
However, three goals in the space of 13 minutes left the home fans jubilant at the final whistle.
Luton kicked off attacking the River End goal and went ahead in the 15th minute when Vine beat Paul Gallacher with a right-foot drive from 25 yards.
The Canaries had enjoyed most of the ball up to that point, but the goal seemed to take the wind out of their wings.
When Norwich did attack Darren Huckerby squared the ball to Robert Earnshaw, who unselfishly left it for Carl Robinson. Unfortunately for him the ball was cleared off his toe.
Earnshaw then attempted to lob Marlon Beresford but cleared the crossbar as well as the visiting goalkeeper.
In the 32nd minute Gary Doherty got on the end of Lee Croft's right wing cross but Beresford made a smart catch. And within seconds the Luton keeper had to palm away Dickson Etuhu's low shot, before Earnshaw's snapshot went a foot wide.
Seven minutes after the break, Morgan made it 2-0 when he beat Gallacher with an angled drive from seven yards out, the ball finding the net at the far post.
Five minutes later Gallacher's goal-kick was picked up by Huckerby, who carried the ball in to the Luton penalty area before sliding a right-foot shot past Beresford from 15 yards.
In the 67th minute City were on level terms. Croft latched on to Carl Robinson's throughball and beat Beresford with an angled drive from the edge of the six-yard box.
The Canaries then took the lead in dramatic fashion. Huckerby's shot came back off the post, Etuhu's follow up was blocked by Beresford, but he could do nothing to keep out Earnshaw's low shot from 6 yards.
To their credit Luton came back at Norwich, who had Gallacher to thank in stoppage-time for an acrobatic save to keep out Warren Feeney's well-struck volley.