Former England captain Karen Walker doubled the lead six minutes before half-time and England winger Sue Smith completed the victory with the third goal 22 minutes into the second half.
Midfielder Sally Lacey scored Birmingham's consolation goal in the last minute.
Charlton Athletic also reached the semi-finals of the competition with an emphatic 5-0 win at Bristol Academy on Sunday.
Liverpool went through on penalties following a 2-2 draw after extra time at Newcastle and now meet Leeds in the semis.
Arsenal's quarter-final at Chelsea was postponed due to a waterlogged pitch.