The visitors were ahead after five minutes when debutant goalkeeper Stuart Moore made a hash of his punch and Will Boyle scored from close range.
Isaac Vassell missed an open goal from five yards for the hosts and Jonathan Smith's header was then cleared off the line by Jack Barthram.
The hosts again came close to a leveller midway through the half as Smith's half-volley went close, while Jordan Cook's wonderful free-kick cannoned against the angle of post and bar.
However, the Robins broke with devastating pace to make it 2-0 on 28 minutes, with Carl Winchester picking out Barthram to fire past Moore.
Luton did reduce the deficit on 41 minutes, Vassell slamming home from inside the box.
Cheltenham had a third on the hour, with Billy Waters on hand to beat Moore and score his 12th of the season.
The Hatters were back in the game once more in the 72nd minute when Liam Davis turned Olly Lee's drilled free-kick into his own net, but the hosts couldn't find an equaliser.
Match report supplied by the Press Association
Match ends, Luton Town 2, Cheltenham Town 3.
Second Half ends, Luton Town 2, Cheltenham Town 3.
Attempt missed. Scott Cuthbert (Luton Town) right footed shot from the centre of the box is just a bit too high.
James Justin (Luton Town) is shown the yellow card for a bad foul.
Foul by James Justin (Luton Town).
Kyle Storer (Cheltenham Town) wins a free kick in the attacking half.
Foul by Glen Rea (Luton Town).
Daniel Wright (Cheltenham Town) wins a free kick in the attacking half.
Attempt blocked. Jordan Cook (Luton Town) right footed shot from the centre of the box is blocked.
Substitution, Cheltenham Town. Kyle Storer replaces Jordan Cranston because of an injury.
Delay in match Jordan Cranston (Cheltenham Town) because of an injury.
Attempt blocked. Jordan Cook (Luton Town) right footed shot from outside the box is blocked.
Attempt missed. Liam Davis (Cheltenham Town) right footed shot from outside the box is just a bit too high.
Attempt missed. Manny Onariase (Cheltenham Town) header from very close range is just a bit too high.
Corner,  Cheltenham Town. Conceded by Scott Cuthbert.
Foul by Scott Cuthbert (Luton Town).
James Rowe (Cheltenham Town) wins a free kick in the defensive half.
Jordan Cook (Luton Town) wins a free kick in the defensive half.
Foul by Manny Onariase (Cheltenham Town).
Foul by Pelly Ruddock (Luton Town).
Harry Pell (Cheltenham Town) wins a free kick in the defensive half.
Substitution, Cheltenham Town. Daniel Wright replaces Kyle Wootton.
Own Goal by Liam Davis, Cheltenham Town.  Luton Town 2, Cheltenham Town 3.
William Boyle (Cheltenham Town) is shown the yellow card for a bad foul.
Jordan Cook (Luton Town) wins a free kick in the defensive half.
Foul by William Boyle (Cheltenham Town).
Substitution, Luton Town. Jake Gray replaces Luke Gambin.
Delay in match Harry Pell (Cheltenham Town) because of an injury.
Foul by Olly Lee (Luton Town).
Harry Pell (Cheltenham Town) wins a free kick in the attacking half.
James Justin (Luton Town) wins a free kick in the attacking half.
Foul by Liam Davis (Cheltenham Town).
Foul by Isaac Vassell (Luton Town).
Daniel O'Shaughnessy (Cheltenham Town) wins a free kick in the defensive half.
Attempt blocked. James Justin (Luton Town) right footed shot from the right side of the box is blocked.
Foul by Lawson D'Ath (Luton Town).
Harry Pell (Cheltenham Town) wins a free kick in the attacking half.
Substitution, Luton Town. Lawson D'Ath replaces Jack Marriott.
Goal!  Luton Town 1, Cheltenham Town 3. Billy Waters (Cheltenham Town) right footed shot from the centre of the box to the top right corner. Assisted by Kyle Wootton.
Attempt blocked. Jack Marriott (Luton Town) right footed shot from outside the box is blocked.
Struggling Cheltenham produced a superb 3-2 win to stun League Two promotion-chasers Luton Town at Kenilworth Road.