Ryan Christie opened the scoring in the 11th minute, netting from close range after his penalty was initially saved.
Ross County then hit back in the 24th minute, as an unmarked Ross Stewart headed in the equaliser from a curling free kick.
They never threatened after their first goal though, as Celtic regained their composure and took the lead again in the 38th minute as Christie latched on to a precise through ball from Kristoffer Ajer to steer home his second of the game.
It was more of the same from the league leaders after the half-time break, as the Celtic attack put together a series of passes to tee up Socceroo Rogic, who tucked the ball in the bottom corner via a deflection for his first goal of the Premiership campaign.
The result was put beyond doubt shortly after as Michael Johnston embarked on a weaving run down the left side and slotted calmly past the keeper to cap off a dominant performance by Celtic.
The win takes Celtic to 37 points and equal first place with arch-rivals Rangers on the Scottish Premiership ladder. Their next match sees them take on Hamilton on Thursday (AEST).
