Tony Stokes struck in either half for the Hoops, while John Guidetti, Callum McGregor and Jason Denayer all netted in the opening 35 minutes as Celtic produced their best domestic performance of the campaign.
Guidetti (11) hit the opener when Stokes sent a free-kick into the County penalty area and the ball spun off a defender in an aerial challenge before dropping for the Swede to volley under home keeper Mark Brown from inside the six-yard box.
County were opened up again three minutes later when Stefan Johansen lifted a diagonal pass to Guidetti on the right side of the box and the striker cut the ball back for the arriving McGregor to score with a low drive.
Johansen was also pivotal in the third goal on 29 minutes as he drove forward and released the ball to Stokes, who stepped on to the pass and curled his shot inside the post from the edge of the box.
Six minutes later the game was finished as a contest when Brown denied Mikael Lustig with a save at his back post from a Johansen corner before the ball spun around the six-yard box where Denayer scooped his effort into the roof of the net.
Adam Matthews, who replaced the injured Emilio Izaguirre just after half-time, laid a pass inside to Stokes (56) and the Republic of Ireland international took three steps forward and crashed the ball in off Brown’s right post from 25 yards.
The home side tried to rally quickly after conceding the two early strikes and Celtic goalkeeper Craig Gordon had to be alert to punch away a fierce free-kick from Richard Brittain at his near post.
But Celtic almost struck again on 26 minutes when Johansen set up Guidetti with a sublime reverse pass and this time he clipped the ball over Brown but just wide of the far post.
Man-of-the-match Johansen was denied a deserved goal in the final minute of the opening half when Guidetti took a cut-back from Izaguirre and rolled a pass to the Celtic midfielder to fire past Brown, but Scott Boyd was back on the line to head clear.
Celtic lost Izaguirre just after the break when the left-back was caught by a heavy Michael Gardyne challenge and had to be carried off with possible ankle ligament damage.
County’s best chances came in the final 25 minutes when Joe Cardle curled his low shot a foot wide from the edge of the box, and Gardyne later thumped the post from a similar range.
McGregor should have added a sixth 12 minutes from time when Scott Brown prodded the ball through to the winger, who raced into the box but County keeper Brown deflected his shot wide with his left leg.
And Celtic skipper Brown played a first-time wall-pass to Stefan Scepovic who sent his low drove from 18 yards inches past the post.