John Fitzgerald gave the Shakers a 34th-minute lead but Chris Casper's side could not hold on in the second period, as Kelvin Langmead netted and Ben Davies slammed home from the spot late on.
Bury had an early Jon-Paul Pittman goal disallowed before Langmead hit the outside of the post with an acrobatic volley for the visitors.
The hosts gradually seized the initiative and Andy Bishop went close before Pittman again got round the back of the Shrewsbury defence before steering his shot just wide of goal.
Eleven minutes before the interval Fitzgerald headed home Brian Barry-Murphy's centre to give the Shakers a deserved lead.
On the stroke of half-time Barry-Murphy brought down Langmead in the area but Sagi Burton was unable to beat Alan Fettis from the spot.
It looked like Bury would go on to double their lead with Marc Pugh and Pittman causing havoc on the wings, but Chris Casper's side struggled to find the final ball to kill off the Shrews.
The game ebbed and flowed in the second period and the visitors got back into it just after the hour mark.
Derek Asamoah got down the outside of Tom Kennedy and delivered a pinpoint cross for Langmead to steer the ball in at the far post.
It was perhaps more than the Shrews deserved, but was indicative of Bury's luck so far this season.
The comeback was completed when Davies was felled by Dave Challinor, who was lucky to avoid a red card.
Ex-Chester City man Davies picked himself up and slammed the ball into the roof of the net past Fettis for the winning goal.