Mancini did get off to a strong start with a 2-2 draw against Juventus in the Champions League on Wednesday, but was unable to follow that up in his opening Super Lig match.
Gala quickly went 2-0 down to Akhisar, who sit third in the Turkish league after a fine start to the season.
Didier Drogba pulled one back for the visitors 20 minutes from time, but Gala were unable to find a late equaliser.
The defending champions have now won just one of their first six league games and sit 13th in the table.
That poor run ultimately cost Fatih Terim his job last week, with Mancini quickly brought in as his replacement on a lucrative contract.