Bjorn Runstrom's second Luton goal in the final game of his current loan spell from Fulham deservedly put the hosts ahead only for Chris Martin to make it 1-1 at the break.
Drew Talbot then got his second Town goal as Hatters dictated only for Jason Shackell to level late on before Simon Lappin curled home a free-kick in injury time.
The game kicked off 15 minutes late after an electrical fault left the Oak Road End in darkness.
But once the action began, Luton almost lit up the match within 45 seconds as Paul Gallacher spilled Matthew Spring's 25-yarder after Kevin Foley's lay-off only for Adam Drury to clear.
The Hatters looked pumped up and went ahead in the 11th minute as David Bell bundled his way past three challenges on the left and squared for Talbot to swipe thin air before Runstrom swept home from six yards.
Fluid-looking Luton came again and only the finger-tips of Gallacher kept out a rasping 35-yarder from Spring as the ball thudded against the stanchion and went out for a throw-in.
Town looked in control until Spring wastefully gifted possession to Lee Croft, who instantly found Martin to roll home into the far corner from 16 yards for a 26th minute equaliser out of nothing.
The visitors came back into it and only a quick intervention from Foley stopped Martin getting a second six minutes later after Shackell had nodded down a Youssef Safri free-kick.
Town started the second period like they did the first and went ahead again three minutes in when Bell, who had looked menacing all evening, latched on to Leon Barnett's pass down the left, beat the committed Gallacher and crossed for Talbot to nod into the empty net from eight yards.
Talbot almost got in again when a Keith Keane cross caused panic in the Norwich box just before the hour.
And the nerves appeared to spread for the visitors when shaky-looking Gallacher almost dropped a Bell 20-yarder into his net in the 63rd minute only to grab the ball on the line at the second attempt.
Luton again appeared in control only to concede out of the blue once more in the 73rd minute when Shackell was left unmarked to nod in Safri's free-kick from the left at the near post.
Darren Huckerby fired in a 77th minute 25-yarder that Marlon Beresford blocked before Spring then worked Gallacher with a 25-yard screamer four minutes later.
Gary Doherty had a header cleared off the line in the final minute, but Lappin ended up the hero for Norwich by curling home a 20-yard free-kick beyond the reach of Beresford and into the net.