Gary Hooper netted from close range after a 14th-minute corner before Scott Boyd’s own goal early in the second half removed any doubt about the outcome of the game.
Celtic survived one tense moment in between the goals when Michael Gardyne volleyed against the post but County otherwise did not look like emulating their 2010 Scottish Cup semi-final win over Neil Lennon’s men amid torrential rain.
Celtic are much changed from the team that suffered arguably the biggest shock in the club’s history – none of the players who featured at Victoria Park played at Hampden – but they went into the game under pressure following consecutive defeats by Atletico Madrid and Rangers.
Lennon admitted he made selection errors for the 4-2 loss at Ibrox and the Celtic manager made five changes tonight.
Mark Wilson, Georgios Samaras, Scott Brown and Ki Sung-yueng all missed out through injury, with Wilson also suspended, while Glenn Loovens dropped out of the squad.
Adam Matthews, Daniel Majstorovic, Joe Ledley, James Forrest and Anthony Stokes came in.
There was no place in the 16 for Kris Commons, who Lennon had stated was in contention to play.
Commons, who was at Victoria Park, posted on Twitter on Sunday that he had been available to face Rangers but Lennon later insisted he had been left out of the squad for Ibrox because of a tight groin.
While the only stripped member of the Celtic squad to have played in that Hampden defeat against County was substitute goalkeeper Lukasz Zaluska, six of the home team played that day.
But Celtic quickly settled their nerves after a scrappy start when Hooper netted.
Majstorovic was close to connecting with Charlie Mulgrew’s inswinging delivery and Stokes retrieved the ball beyond the far post before clipping a cross back in.
Ledley rose to head towards goal and Hooper tapped the ball over the line from two yards in a crowded six-yard box.
Celtic looked to build on their lead and Stokes had a first-time effort deflected over before some desperate defending helped the hosts survive Mulgrew’s corner.
But Celtic survived a scare in the 20th minute when Kelvin Wilson’s weak headed clearance was met full on the volley by Gardyne, whose 22-yard strike bounced off the post.
The hosts pushed Celtic back for a spell without carving out a chance and Lennon’s side came back into it before the break.
Forrest’s shot was blocked by Grant Munro before Celtic cut open their opponents through some one-touch passes from Forrest and then Hooper to set Ledley through.
However, Michael Fraser narrowed the angle and blocked the Welshman’s shot with his trailing foot.
County had the final chance of the half though when Badr El Kaddouri failed to clear and Rocco Quinn sent the ball low into the box, but Colin McMenamin could not connect properly with his volley and Fraser Forster made a comfortable save.
Wilson, who needed treatment to his lower leg after a foul by McMenamin, did not come out for the second half.
Victor Wanyama came on in central midfield with Mulgrew moving to centre-back and Ledley switching to the left.
Celtic moved two ahead in the 52nd minute. Fraser held Hooper’s 20-yard shot at the second attempt and gave the ball to Iain Vigurs, but the midfielder was caught in possession deep inside his own half.
Celtic eventually worked the ball to Stokes on the right and he sent in a driven cross that was turned home by Boyd without any pressure from a visiting player.
Boyd soon cleared a similar cross from Stokes as Celtic broke and Fraser blocked Hooper’s shot after a Ledley knockdown.
Although the hosts did not drop their efforts, the game petered out in the final quarter.
Irish youngster Paul George almost grabbed his first goal for Celtic moments after coming on when he turned Ledley’s low cross towards goal in the closing stages, but Fraser dived to make a brilliant stop.