Simon Lappin hit a brilliant injury-time free-kick to seal a vital win for Norwich and leave Luton deep in Coca-Cola Championship relegation trouble.
The match looked to be heading for a draw before Lappin curled home from 20 yards in the 93rd minute to move Norwich six points clear of the bottom three.
Earlier, Luton had twice taken the lead through Bjorn Runstrom and Drew Talbot, with the Canaries equalising on both occasions thanks to Chris Martin and Jason Shackell
The kick-off was delayed by 15 minutes to 8pm because of power problems which prevented fans from entering the stadium through the turnstiles.
After four minutes, Norwich suffered an injury scare when defender Adam Drury fell awkwardly over an advertising hoarding after winning a corner for his team.
Play was held up for almost three minutes so Drury could have treatment before he eventually rejoined the action.
Matters became even worse for the visitors when Luton took the lead in the 11th minute.
The excellent David Bell tricked his way past former Hatter Gary Doherty inside the box before cutting the ball back into the danger zone, where Talbot's clever feint allowed Runstrom - on loan from Fulham - to convert from six yards.
The goal preceded a short spell of Luton pressure, and Matthew Spring was unlucky to see a blistering 35-yard drive cannon back off the crossbar.
After that near miss, Norwich found their way back into the match, equalising in the 26th minute.
Following a mis-placed pass from Spring, the ball found its way to Martin, who composed himself before finishing with an impressive angled shot from 15 yards.
In stoppage time, Martin steered wide from seven yards after Darren Huckerby had beaten Markus Heikkinen on the left before setting up his strike partner.
The Canaries ended the first-half with the upper hand but it was Luton who regained the lead three minutes after the break.
Bell took advantage of Andy Hughes' slip to round goalkeeper Paul Gallacher and lift the ball back into the middle for Talbot to head into the empty net from six yards.
Nine minutes later, Talbot should have made the game safe for Luton after a defensive mix-up, but tried to take the ball past Gallacher instead of shooting first time, and the chance was lost.
At the other end, Huckerby's back-heel set up Lee Croft, only for Lewis Emanuel to throw himself in the path of the winger's low shot.
Luton looked relatively comfortable, but they were stung in the 73rd minute when Youssef Safri's inswinging free-kick from the left was headed in at the near post by Shackell.
Two minutes after scoring, Shackell rescued his team at the other end by blocking Steve Robinson's shot from 18 yards.
Then, nine minutes from full-time, Gallacher denied Spring from 25 yards with an acrobatic save to concede a corner before Spring fouled Croft in stoppage time and Lappin scored from the free-kick.
Teams:
Luton Beresford, Foley, Heikkinen, Barnett, Emanuel, Spring,Keane, Robinson, Bell, Runstrom (Feeney 85), Talbot.
Subs Not Used: Brill, Morgan, O'Leary, Boyd.
Goals: Runstrom 11, Talbot 48.
Norwich Gallacher, Drury, Shackell, Doherty, Croft, Safri,Hughes, Lappin, Etuhu (Fotheringham 64), Huckerby, Martin.
Subs Not Used: Lewis, McVeigh, Colin, Eagle.
Booked: Etuhu.
Goals: Martin 26, Shackell 73, Lappin 90.
Att: 8,868
Ref: U Rennie (S Yorkshire).