Rogic, who has signed a new three-year contract extension with Celtic, produced a man-of-the-match performance to help Celtic progress to the quarter-finals.
In the 19th minute the former Central Coast Mariners star finished off a wonderful team move with a classy turn and volley.
Rogic was involved again in the 60th minute, providing an assist by laying off the ball to Scott Sinclair who curled his shot into the top corner to make it 3-0.
With 15 minutes remaining, Rogic scored his second and Celtic's fifth goal when he tapped home from close range after Motherwell's goalkeeper, Craig Samson, could only parry James Forrest's shot.
Earlier in the match Sinclair had won a penalty, which was converted by Moussa Dembele, to make it 2-0. Dembele was involved again in the 64th minute when his deflected shot made it 4-0 to the The Hoops.
Celtic will now face Alloa Athletic in the quarter-finals which are scheduled for late September.