The Hatters had taken the lead nine minutes before half-time through Asa Hall's seventh of the season.
But a super show from the Shakers after the break saw the visitors claim all three points to keep their League Two promotion push on track.
In a slow opening to the game it was Luton who fashioned the first opening in the eighth minute when full-back Michael Spillane snatched at a 10-yard volley sending the ball wide after Kevin Nicholls' corner had only been half cleared.
Two minutes later Bury went close as captain Paul Scott stole in at the far post after corner only to see his shot cleared off the line by Sol Davis.
There was little goalmouth action of note until the 25th minute when Tom Craddock, making his first start since his permanent move from Middlesbrough, cracked a super 16-yard left-foot volley straight at visiting goalkeeper Mark Tyler after Nicholls' corner had fallen into his path.
Bury had been quiet until this point but came alive in the 26th minute when a swift break saw Elliott Bennett cross for Bishop to round Conrad Logan only to see his shot blocked by a combination of Kevin Gallen and Spillane.
Sixty seconds later the Shakers went closer still when Morrrell side-stepped Ed Asafu-Adjaye and curled a 22-yard shot that bounced away off the far post.
Bishop then tested Logan three times in the space of four minutes as the visitors turned the screw, but surprisingly it was Luton who took the lead in the 36th minute.
Craddock tried his luck cutting in from the left only to be tackled by Scott, however, the challenge saw the ball roll perfectly for Hall to arrow home his third goal in three games with a terrific low 20-yard strike into the bottom corner.
Luton could have gone 2-0 up four minutes after the interval but Craddock placed his free header into the arms of Tyler from 12 yards after Gallen's cross.
Instead it was 1-1 in the 59th minute when Elliott Bennett made ground down the right and sent in a cross that deflected off the unfortunate Hall over Logan and into the net.
Bishop followed up to make sure the ball was over the line and ran off claiming the goal.
Bennett was involved again in the 66th minute as Bury took the lead. The winger beat Asafu-Adjaye to the ball, this time on the left flank, and crossed for Morrell to dive and head home at the near post.
Luton heads dropped and Bury finished the game in complete control.
Chances came and went with the best falling to Bishop in the 84th minute when he rounded Logan only to see his shot impressively cleared off the line by Mark Bower.