The Australian's controlled half-volley in the 23rd minute at Pittodrie was enough to send Celtic 10 points clear of Aberdeen with a game on hand on the team who have finished runners-up in the past two seasons.
Celtic passed up a stream of chances either side of the interval and defended well when Aberdeen pushed for an equaliser in the final quarter.
All but two of the eight players who came in for Celtic's 4-0 victory over Ross County dropped out of the team
Stuart Armstrong and Emilio Izaguirre were the exceptions, the latter after Kieran Tierney suffered a training-ground ankle injury that will keep him out for two months.
Aberdeen gave captain Ryan Jack his first start in 10 weeks following knee surgery while top goalscorer Adam Rooney and James Maddison started on the bench, Wes Burns being preferred up front.
Erik Sviatchenko was booked in the first minute for chopping down Jonny Hayes as Aberdeen started on the front foot, but the closest they got to an early goal was when the winger had a penalty-box effort blocked by Jozo Simunovic.
Celtic grew in possession but they had not seriously threatened before opening the scoring.
Mikael Lustig found room to cross and the ball was only half-cleared to Rogic on the edge of the box
The Australian took one touch with his right foot and quickly sliced across the ball with the outside of his left foot to send it into the corner of the net.
Aberdeen took time to respond but Niall McGinn twice found Andrew Considine in good positions with crosses
The left-back headed the first over, under pressure, before his volley was blocked.
Celtic had a great opportunity to double their lead just before the break but Armstrong shot wide from 12 yards after being set up by James Forrest.
Celtic maintained their pressure after the interval and Joe Lewis produced a brilliant fingertip save to stop Scott Sinclair's effort curling in the top corner after the wide player cut inside.
After making small inroads at the start of the second period, Aberdeen brought on Rooney and Maddison in the 53rd minute but they were soon caught out by Lustig's diagonal ball, Lewis again coming to the rescue to deny Rogic.
The goalkeeper again kept the game alive when he saved from Moussa Dembele and Rogic, before the Australian and Forrest both shot wide from 18 yards.
Maddison was starting to pose questions to Celtic and his free-kick found Rooney but the striker's backwards header was cleared in the goalmouth.
A melee erupted when Scott Brown kicked the ball against the grounded Graeme Shinnie, who had been decked by Lustig
The Celtic pair were booked along with Hayes, who pushed the visiting captain over.
Aberdeen made a late surge for an equaliser but Sviatchenko produced a last-ditch sliding tackle to deny Shinnie and Maddison volleyed just over from 16 yards.
Armstrong shot just over at the other end but the game finished in the Celtic penalty box with Lewis among those attacking several testing set-pieces from Maddison, and Craig Gordon saved brilliantly from Sviatchenko's misplaced header five minutes into injury-time.