Green was beaten by a speculative 35-yard Carlos Edwards cross and had to pick the ball out of the net a further three times after goals from Warren Feeney, Peter Holmes and Steve Howard as Town ran riot.
The second half was a different story as Dean Ashton and Ryan Jarvis reduced the arrears, but by then Hatters had done more than enough to humiliate one of last season's Premiership sides.
To compound Norwich's misery, on-loan defender Calum Davenport was sent off in the 81st minute for a second bookable offence.
Town lined up without their influential skipper Kevin Nicholls who missed his first league and cup game of the season with a foot problem.
He was replaced by Steve Robinson, who returned from a two game absence after a heel injury. Norwich included ex-Hatter Gary Doherty in their starting 11.
The game kicked off with the visiting Norwich fans in fine voice, but they were almost silenced just four minutes in when Dean Morgan curled a 25-yard free-kick just past the upright.
A superb Edwards tackle on Darren Huckerby then set Luton on a slick counter-attack two minutes later that saw Howard head Ahmet Brkovic's hanging cross over the bar.
The visitors soon settled though and when Edwards completely missed Ashton's knock down in the ninth minute, Huckerby was given a clear sight on goal only to pull his left-footed shot wide from 12 yards.
Town survived another let-off two minutes later when Marlon Beresford could only tip Jim Brennan's deep cross into the path of Ian Henderson who curled a 15-yard effort against the bar. From the follow-up, Huckerby could only blast wildly over.
However, it was Hatters who took the lead in the 14th minute when stand-in captain Chris Coyne saw a volley deflect out to Howard on the right who crossed low and hard at the second time of asking for Feeney to prod the ball high into the net at the near post for his fourth of the season.
The hosts looked comfortable after the goal until Ashton stepped up to slam a 22nd minute free-kick that deflected off the wall and just wide from fully 40-yards.
But if that was close to being a freak goal, that was certainly the description for Luton's second on 27 minutes.
Edwards looked to have wasted a promising Town move when he overhit Holmes' lay-off from way out on the right flank, but as Norwich goalkeeper Green followed the flight of the ball he inexplicably slipped, allowing the ball to bounce into the net.
With chants of 'Scotland's number one' ringing round the ground, Green was almost embarrassed again as Robinson came close to robbing the goalkeeper after a back pass.
Green did redeem himself with a fine stop from Holmes' stinging 16-yard volley in the 33rd minute before being beaten for a third time a minute later as Holmes played a one-two with Feeney and then fired home low into the corner from 18 yards.
With Norwich in complete tatters, Luton grabbed a fourth four minutes before the break as Holmes walked past Simon Charlton in midfield and crossed for Howard to nod home his sixth of the season at the far post.
Norwich finally came back into the game in the 56th minute when, with Sol Davis off the field receiving treatment, Ashton rifled home a low strike from 22 yards.
Davenport then saw red for two bookable offences in the space of 12 minutes before substitute Jarvis headed home Ashton's cross five minutes from time.