The Reds had come from behind to lead 2-1 at Prenton Park, with Curtis Jones scoring 20 minutes from time and Toni Gomes seemingly clinching the victory in extra-time.
But the loss of Liam Coyle to a second yellow card late on turned the game back in the visitors' favour, and they levelled things up through Kian Flanagan to send the game to penalties.
That late comeback proved immaterial, however, with Kamil Grabara producing two stops in the shootout to mark himself out as the matchwinner and set up a fourth-round clash with Manchester City in the new year.