Virgil van Dijk scored his first goals for the league leaders and Joe Ledley netted the other two after Ivan Sproule had halved the advantage.
Celtic dominated the opening period but lacked a cutting edge until Charlie Mulgrew and Emilio Izaguirre linked at the edge of the County penalty area and the latter crossed to the back post where van Dijk (41) cushioned his header in off the woodwork.
And the central defender struck again on 53 minutes when he ghosted into the six-yard box to glance another header into the net from an inswinging Mulgrew corner.
Ross County substitute Sproule (68) gave the hosts some hope when he rifled home from close range after Fraser Forster did well to keep out Rocco Quinn’s deflected shot.
But the Hoops restored the two-goal advantage just two minutes later when Ledley sent Brown the wrong way after the midfielder rose to meet an Izaguirre chip.
The Wales midfielder completed his brace on 73 when he glanced yet another header past Brown, this time from a Darnell Fisher delivery.
Ledley was denied a hat-trick by the post after Brown had got down to parry a powerful Tom Rogic shot.
Brown made an impressive diving save on just eight minutes when van Dijk weaved past a couple of challenges and let fly through the legs of Brian McLean from 25 yards.
Teemu Pukki’s link play was impressive from the start of the match, and he laid a weighted pass to Izaguirre to send a first-time cross to the edge of the box for James Forrest to lift the ball over the bar.
And Celtic almost struck a quick second goal before the break when Branislav Micic blocked Ledley’s drive on the six-yard line and the rebound fell to Nir Biton 25 yards out, but his low shot was well held by Brown.
Van Dijk and Scott Brown both sent efforts over the bar from outside the penalty area in the early stages of the second half before the Celtic skipper had an angled shot deflected round the post by the County keeper’s foot moments before the second goal.