The Hatters' cause was not helped by an amazing fluke goal for the Tigers, a disallowed goal and a poor display by the referee Trevor Parkes, who harshly sent off their captain Kevin Nicholls midway through the second half.
But keeper Dino Seremet's shaky performance seemed to filter through to the rest of the team, who never looked like recovering after they gifted City three goals.
Luton did have the brighter start, though, and in the 10th minute Boaz Myhill did well to acrobatically tip over a flicked header by Steven O'Leary.
But barely two minutes later, Stuart Elliott stunned the large and vocal away following when he raced away to the left and spotted Seremet off his line. He unleashed a cross-cum-shot that appeared to be drifting out of play, but the ball ended up in the top right-hand corner of the net.
Suddenly it was the City faithful who were in good voice.
Elliott nearly grabbed his second with a low close-range header in the 15th minute but for a brilliant reflex save from Seremet, but he was then harshly booked for fouling the Hatters keeper.
His counterpart Myhill was called into action again just a minute later, however, when he beat a speculative long-range shot by Steve Howard over the bar.
But in the 23rd minute, Luton committed defensive suicide again when a terrible mix-up between Seremet and Paul Underwood saw the ball roll towards an open goal, with Delroy Facey chasing and, despite a desperate bid to haul it off the line, Facey's effort could not be kept out.
Luton's Kevin Foley had the ball in the net in the 36th minute with a decent curling shot, but his effort was controversially ruled offside.
Just seconds later, however, Elliott made it three after yet another calamity in Luton's box - Seremet failed to catch what looked like an easy cross, and City's Northern Ireland international was there to bundle in the rebound from a few yards out.
There was even more woe for Luton when Nicholls was given a straight red card in the 62nd minute for a challenge on Elliott, which, although it looked like he was struck in the face, seemed a little harsh.
In the 67th minute, Rowan Vine's long-range effort was easily dealt with by Myhill, but it all seemed too little, too late, and City were completely in control - for perhaps the first time all season.
This was evident in the 75th minute when Facey was allowed time and space to cleverly tee up the ball for Green, but he skewed his shot disappointingly wide, while 10 minutes from time, Facey then tested Seremet twice in a minute.
City kept up the pressure until the very end, hardly giving Luton a look-in.
Substitute Michael Keane brought another good save from Seremet in the 85th minute, and two minutes later was denied again as Seremet somehow saved his team from losing by a wider margin.