Goals from Tom Craddock and Ian Roper in the last ten minutes of the first half put the visitors in charge.
And although Andy Bishop hit back with a 53rd minute penalty, the Shakers failed to avoid a third straight home defeat.
Bury boss Alan Knill changed his starting line-up for the first time in 11 league games following the midweek home defeat by Rotherham with Ben Futcher and Andy Morrell making way for Ryan Cresswell and on-loan striker Chris O'Grady.
Luton made a lively start with Chris Martin firing into the side-netting after being played in by a quickly-taken free-kick from Jake Howells.
O'Grady hit the top of the crossbar in the sixth minute after lobbing Luton goalkeeper Conrad Logan and the visitors had a goal disallowed shortly after when Howells' goal-bound effort went in off Martin, who was prone on the goal-line.
O'Grady then fired wide from just outside the box, while Logan made a fine low save to turn away an Bishop header.
But Luton broke the deadlock shortly after when Bury keeper Wayne Brown somehow let Craddock's 12-yard strike squirm from his grasp and over the line.
Logan did well to deny Bury defender Cresswell an equaliser and the Hatters doubled their lead just before the break.
Bury full-back David Buchanan failed to deal with a corner and Craddock crossed for the unmarked Roper to head home.
Richie Baker replaced Mike Jones at half-time and within eight minutes of the restart was brought down by Roper in the Luton box.
Bishop, who missed from the spot in midweek, converted with aplomb, but for all their possession following that goal, Bury could not create any clear-cut chances.