Mouche, who had been booked earlier in the game, scored the only goal as he broke down the right onto a flick-on, held off the challenge and tucked in, before steaming down the touchline with his shirt off. He was then dismissed.
The result leaves Gala five points behind Fenerbache, which has two games in hand, and comes in the wake of its dismal UEFA Champions League loss at Chelsea.
Theofanis Gekas scored twice as Konyaspor claimed a 2-1 win over Bursaspor despite being reduced to 10 men.
Gekas scored in the 66th and 73rd minutes, with Erdinc Yavuz seeing red for the visitor in the 90th minute for his second bookable offence. Substitute Enes Unal scored a last-minute consolation for Bursaspor.
Elsewhere, Antalyaspor drew 0-0 at home to Karabukspor while Caykur Rizespor and Eskisehirspor's match also finished goalless, with the latter's Jerry Akaminko sent off in the 83rd minute.
