Stuart Armstrong opened the scoring for the champions in the 17th minute, gaining possession inside the area before cutting inside and firing beyond Accies goalkeeper Gary Woods into the bottom corner.
The Frenchman was eventually dispossessed inside the box, but the ball fell kindly to the in-form Scott Sinclair, who fired into the top corner.
Sinclair netted his second and Celtic's third of the night just minutes before half time when he was in the right place to tuck home a rebound, putting Rodgers's side almost out of sight at the break.
Edouard marked his Hoops debut with a goal just after the hour mark, using his pace and strength to carve out space in the box before shooting low into the bottom corner.
Hamilton netted a late consolation when Alex Gogic planted a free header into the back of the net, but it counted for little in the end.