Selcuk Inan's 38th-minute penalty proved the difference between the two sides and means Roberto Mancini's men now occupy the all-important second UEFA Champions League qualifying berth.
In extending their unbeaten league run to 11 matches, Gala climbed two points above Besiktas, but remain four points behind Fenerbahce, who needed a late equaliser to earn a 1-1 draw at Elazigspor.
The hosts went in front on the stroke of half-time through Onur Ayik and Fenerbahce looked set for a third defeat in four when Cristian Baroni was sent off just before the hour mark.
However, Mehmet Topal struck seven minutes from time to rescue a valuable point for the leaders.
Fourth-placed Sivasspor suffered a 6-2 thrashing at the hands of Kasimpasa, who moved up to sixth, three points behind their opponents.
First-half goals from Ezequiel Scarione, Sanharib Malki and Ryan Babel put Kasimpasa in control and Tabare Viudez made it four just after the break.
The visitors replied through Hakan Arslan and Aatif Chahechouhe, either side of Tunay Torun's strike, before Andre Castro's goal and the dismissal of Manuel da Costa capped a miserable day for Sivasspor.
At the wrong end of the table, Kayseri Erciyesspor claimed a second successive win with a 2-1 home victory over Karabukspor.
Goals from Murat Yildirim and Yasin Oztekin put the home side in control and the hosts held on to move within four points of safety despite Murat Akca scoring to set up a nervy final 15 minutes.
Trabzonspor showed no ill effects from their midweek UEFA Europa League defeat to Juventus, claiming their first win of 2014 with a 2-1 home success over Kayserispor.
They had to come from behind, Srdan Mijailovic giving the visitors the lead before second-half goals from Adrian Mierzejewski and Marc Janko turned things around.
Bursaspor won 2-0 against struggling Rizespor, while Gaziantepspor and Akhisar Belediyespor drew 1-1.
Elsehwere, two Bogdan Stancu goals saw Genclerbirligi come from two down to earn a point in a 2-2 draw with Konyaspor, while the game between Antalyaspor and Eskisehirspor finished goalless.
