A spectacular strike from Rowan Vine had given the Hatters a half time lead before Dean Morgan doubled the visitors' lead on 52 minutes.
With Luton threatening to score a third, Darren Huckerby was sent clear to beat Marlon Beresford. Moments later, the impressive Lee Croft notched his first goal in a yellow shirt when he shot across Beresford.
The comeback was completed when Robert Earnshaw scored his second of the season on 71 minutes. City weathered a late storm from Luton, both sides having had to endure some terrible conditions for much of the game, but hung on for the victory.