The 23-year-old, who joined Abdullah Avci's side from Adana Demirspor last season, gave them the lead on 25th minute, when he rose above his marker to head home a cross from Edin Visca.
Former Manchester City and Real Madrid striker Emmanuel Adebayor made it 2-0 on 42nd minute with a perfect finish before former Swansea City star Bafetimbi Gomis pulled one back for Galatasaray on 55th minute.
However, Adebayor restored Basaksehir's two-goal lead just three minutes later, when he connected a pass from Marcio Mossoro before completing his hat-trick on 76th minute from the spot.
Substitute Kerim Koyunlu Frei completed the demolition on 89th minute for Basaksehir, just three minutes after coming on for Eljero Elia.
The striker was set up by Gael Clichy and he calmly slotted in his last ditch effort past goalkeeper Nestor Fernando Muslera.
Attamah has now scored two goals in 11 appearances, with Basaksehir sitting second on the log, tied on 26 points with Galatasaray but the latter have superior goal difference.