Steve Schumacher scored a last-minute equaliser to deny Shrewsbury their first away win of the season.
The Bury captain headed home Gregg Wylde's 90th-minute corner to seal a 2-2 draw in an exciting battle between two struggling League One sides.
Shrewsbury took the lead within 31 seconds, Aaron Wildig following up after Marvin Morgan had been tackled to hammer the loose ball home.
But Bury hit back when David Worrall fed Matt Doherty and the full-back slotted a low shot under `keeper Chris Weale.
Weale did better with a powerful right-footed shot from Wylde, reacting quickly to deny Bury a second.
Later, Darren Jones got lucky when he deflected a Wylde effort against his own post with Weale stranded.
But Shrewsbury eventually edged back in front after Taylor raced down the right flank to smash an unstoppable shot into the corner of the net with 68 minutes on the clock.
Moments later Morgan missed a sitter, denied by the legs of Trevor Carson when one-on-one.
And his miss proved costly when Schumacher struck at the death to clinch a draw.