All the goals at New Douglas Park were scored during the first half, with hosting Hamilton actually breaking the deadlock in the fourth minute.
Summer signing Gramoz Kurtaj broke into the Celtic penalty area and having been afforded enough space to shoot, the German midfielder got the better of visiting goalkeeper Craig Gordon with a low strike.
However, the goal sparked the away side into life and they drew themselves level with 27 minutes on the clock. Leigh Griffiths found Dedryck Boyata with a lofted cross and the centre-back was able to sweep the ball in.
Then, five minutes later Griffiths got his name on the scoresheet with what proved to be the winning goal. Saidy Janko's cross picked out the Scotland international, who headed into Michael McGovern's net from close range.
Chances were at a premium after the restart, with Celtic's best falling to Kris Commons. However, the winger's effort from the edge of the area was dragged wide.