A Leigh Griffiths penalty after just three minutes set Ronny Deila’s side on their way to an opening-day victory and a Stefan Johansen strike before the half-time break secured all three points.
However, the Hoops boss will be concerned over the fitness of Griffiths ahead of the second leg of their Champions League third-round qualifier against Qarabag in midweek after he limped off just 19 minutes in.
The striker never seemed to fully recover after being hacked down in the box by new County captain Andrew Davies, although he did pick himself up to blast his spot-kick under the diving Scott Fox.
The former Partick Thistle goalkeeper was called into action shortly after to deny James Forrest but there was then a bit of controversy surrounding his opposite number down the other end of the pitch.
Craig Gordon was perhaps fortunate to stay on the pitch after 22 minutes when he came rushing out of his box to foul Jackson Irvine after the midfielder had made a bursting run and rounded the Celtic No 1 before being felled.
Referee Willie Collum showed Gordon a yellow card despite County protests and the Celtic goalkeeper turned Ian McShane’s resulting free-kick behind for a corner.
Gary Mackay-Steven then made up for spurning an earlier chance by laying on a perfectly-weighted pass for Johansen to smash the ball into the corner of the net from 15 yards.
The hosts dominated after the interval, forcing a number of corners, but they had Gordon to thank for keeping out Craig Curran as County looked to get back into the game.
That immediately seemed to spur Celtic back into life as Stuart Armstrong rounded Fox but was denied on the line by McShane.
However, the visitors continued to enjoy a decent spell in the game and had chances with Michael Gardyne firing just wide from the edge of the box before substitute Raffaele De Vita’s powerful effort was palmed away by Gordon.
In an end-to-end second half, Celtic were keen to find a third goal to wrap up the points and substitute Kris Commons nearly provided it on a couple of occasions but he was denied twice by Fox.
Late on Stefan Scepovic, who came on as a first-half substitute for Griffiths, forced Fox into another good save with a driving effort from the right-hand side.