Substitute Gorre's deflected effort flew beyond the despairing Arijanet Muric for his 10th goal of the season.
A cagey start set the tone for much of the first half, but the hosts did threaten through Javairo Dilrosun and Matt Smith.
Rabbi Matondo also came close for City in the the final moments before Gorre struck.
The win takes Swansea, unbeaten in their last seven matches, up to fourth place in Division 1 with 27 points. Man City lie eighth.
Man City: 16 Feb v Sunderland (A) PL2
Swansea: 12 Feb v Newcastle (H) PL Cup