Georgia Stanway's 36th-minute sending off seemed destined to affect the outcome of the game, but City's opening goal only minutes later changed that outlook, Bunny Shaw heading in to give the Blues a first-half advantage.
United came back after the break and found two goals in quick succession - the first a clever flick from Lucy Staniforth, the second an accurate finish from Alessia Russo - to take the lead with only 15 minutes remaining.
Substitute Ellen White drew City back level with a simple two-yard tap-in after Mary Earps had saved a brilliant strike from Jess Parks.