The Reds went ahead against our Championship visitors after seven minutes, when Leah Galton headed us into the lead, before Maya Le Tissier doubled that advantage in first-half stoppage time.
Despite having the majority of the ball after the break, further clear-cut chances were limited and Louise Griffiths set up a tense finish when she pulled one back for the visitors with just over 20 minutes left to play.
The contest looked to be meandering towards a 2-1 finish, but Ella Toone made sure of the victory - and our place in Sunday's semi-final draw - with United's third goal in second-half stoppage time.