Roberto Mancini's side crashed out of the UEFA Champions League on Tuesday with a limp display against Chelsea and Mouche's last-gasp winner saw hosts Gala drop to third in the Super Lig.
While Mouche was sent off for his over-zealous celebrations for rock-bottom Kayserispor,, the strike allowed Besiktas the opportunity to move second - a chance they took by comfortably seeing off Akhisar Belediyespor 3-0.
Veli Kavlak and Mustafa Pektemek scored within the opening nine minute before Atiba Hutchinson added a third midway through the second half as Besiktas took the lead in the race for automatic Champions League qualification.
Besiktas are now the closest challengers to leaders Fenerbahce, after Ersun Yanal's side were comfortable 3-0 winners at 10-man Gaziantepspor in Monday's only game.
Emmanuel Emenike continued his fine season with a brace before Moussa Sow added a third, with Gilles Binya shown a second yellow card late on.
Kayserispor's back-to-back wins will have made life uncomfortable for those immediately above them, although 17th-place Kayseri Erciyesspor were able to pick up a point against UEFA Europa League outsiders Sivasspor.
Edinho's 57th-minute penalty looked to have secured them a fourth win in five, but Aydin Karabulut equalised to send Roberto Carlos' men home with a point.
Fellow strugglers Elazigspor were beaten 3-1 by mid-table Genclerbirligi as they slipped into the drop zone thanks to the goalless draws gained by Rizespor and Antalyaspor.
The duo gained a point each against 10-man Eskisehirspor and Karabukspor respectively, while Konyaspor survived a late scare to see off Bursaspor 2-1.
Theofanis Gekas scored twice for Konyaspor before the hosts pulled a goal back and, while Erdinc Yavuz was dismissed late on, the visitors held on to record their first win in six.
Trabzonspor and Kasimpasa drew 0-0 in Friday's only game to leave the race for the final Europa League spot wide open - with just two points separating ninth and fourth.
