The Shakers slipped out of the top seven on Tuesday but Ryan Lowe claimed the game's only goal in the 39th minute to give Bury their first win in seven and send them above the Shrews into fifth place.
Tom Lees' loose ball gifted Shrewsbury the game's first chance and James Collins was denied by Bury keeper Cameron Belford. Shrews keeper Ben Smith then turned a Steve Schumacher free-kick wide before Bury winger Andy Haworth lifted over from Mike Jones' through- ball.
Jones then volleyed over but Bury took a deserved lead as Jones fired against the crossbar and Lowe bundled the rebound past Smith for his 15th goal of the season.
Smith thwarted Nicky Ajose just after the break but the visitors gradually came into it, with captain Ian Sharps having a shot cleared off the line.
Collins then turned over a Mark Wright free-kick while Mat Sadler fired over from 20 yards. But Jones and Ajose went closest to scoring for Bury late on.