A Mark Hughes own goal looked like handing the promotion-chasing Shrews all the points but Taylor raced through at the death to secure Paul Fairclough's men a valuable point.
The hosts started the stronger and might have taken the lead early on but Clovis Kamdjo's towering header was superbly kept out by Shrews stopper Chris Neal.
The visitors' first opportunity came midway through the first half but Craig Disley fired over the bar when Barnet keeper Liam O'Brien dropped the ball.
But the Shrews made no mistake when Hughes prodded past Neal after a scramble in the Bees penalty box after Jordan Parkes had been robbed of possession.
The visitors were much improved after the break and wasted a number of chances to double their lead - a Mark Wright free-kick fizzed just wide before he shot across the face of goal on 63 minutes. And Shrewsbury were made to pay when Dean Holden squandered possession and Taylor scampered through on Neal before showing nerves of steel to roll the ball home and secure the Bees a point.