Nigerian Enoch Showunmi grabbed their winner, hustling in a near-post effort from another of those accurate crosses from Paul Underwood on the left.
City had a spell of just a quarter of an hour on top, a period sparked by a triple set of substitutions nine minutes into the second half.
The most successful of the changes saw player-manager Brian Tinnion set up a spectacular equaliser for teenager Leroy Lita.
As Tinnion's ball came in from the right it glanced off the head of Paul Heffernan for Lita to throw himself down and score with a shot over his shoulder.
City sensed they might rescue a point they hardly deserved but all their pressure failed to break Luton's calm defence, although Marlon Beresford was forced into one vital save from Heffernan.
City have seldom been so happy to hear a half-time whistle after taking a drubbing from a slick, confident Luton.
They went in one down, a thundering header from Chris Coyne on 34 minutes, but it could have easily have been three or four.
Only the alertness of Steve Phillips in goal kept the score down as the division leaders cut through the home midfield.
Wing back Scott Murray lost out to Underwood inside the touchline on the left to set up the goal.
Underwood's high cross travelled outside the far post where Coyne out-jumped the defenders.
Phillips had earlier saved well at Underwood's feet, and with Showunmi a towering target figure, Luton were not troubled by the absence of Steve Howard sitting out the second game of a three-match suspension.
City, meanwhile, were facing card problems as yellows for Tony Dinning and Steve Brooker could easily have been reds.
Dinning brought down Rowan Vine, clipping his ankles as he made a run on goal, while Brooker was guilty of elbowing Coyne a minute after the defender had scored.
City ended the game with four bookings to Luton's two and Brooker faces a one-match ban for a fifth caution of the season.