With the Reds’ senior players enjoying the inaugural Premier League winter break, under-23s boss Neil Critchley led a side containing seven teen outfield players.
They edged past the Shrews through Ro-Shaun Williams’ own goal.
The Shrewsbury defender headed past his own keeper from Neco Williams’ cross.
At 0-0, Shaun Whalley thought he had given the Shrews the lead however his header was disallowed when a video assistant referee check dominated there had been an offside within the build-up.
Shrewsbury had come back from 2-0 right down to force a fourth-round replay wherever kick-off was delayed by quarter-hour when fans of each clubs were wedged in delays following an accident on the M62.