Swansea came close in the opening minute, with Mumford hitting a free kick against the woodwork.
Roger Freestone saved from Richard Sneekes and Ryan Williams mid-way through the half, while Paul Musselwhite denied Sidibe.
Bradshaw and Alexander also had shots blocked, while late pressure from the home side saw Musselwhite keep out Coates' effort from six yards.
City were unable to continue their good work into the second half, as Swansea got into the game.
Cusack headed over shortly after the restart, and then Musselwhite saved a volley from Sharp.
Alexander shot over on the hour mark, but Swansea were soon on the attack again. Watkin hit the bar with a 70th minute header before finally making the breakthrough 7 minutes from time.
Phillips' left-footed cross found Watkin un-marked, and he headed past Musselwhite to make it 1-0 and that's how it finished.
Starting line-up (4-4-2): Musselwhite; Edwards, Whittle, Greaves, Wicks; Williams (Beresford), Johnsson, Sneekes, Philpott; Alexander, Bradshaw (Whitmore).
Un-used subs: Glennon, Holt, Dudfield.