Celtic booked their place in the quarter-finals of the Scottish Communities League Cup after seeing off the challenge of Ross County.
County, looking to repeat their 2010 Scottish Cup win over Celtic, fell behind when Gary Hooper tapped home on the line after 13 minutes.
The hosts almost levelled when Michael Gardyne's piledriver rattled the post.
Scott Boyd's own goal gave Celtic a healthier lead and Neil Lennon's side were comfortable thereafter.
Celtic join Scottish Premier League contemporaries Dundee United, Hibernian, Kilmarnock, St Mirren, Division One Falkirk and Second Division East Fife in the last eight.
The visitors moved into the lead after Charlie Mulgrew's corner had been cleared and then retrieved by Anthony Stokes.
The striker's cross back into the danger area was headed down by Joe Ledley to the predatory Hooper, who added the final touch.
Celtic were having some success from Mulgrew's deliveries, County taking several attempts to clear after another inswinging corner from the left-sided player.
County's fans were on their feet moments later, only to have their celebrations cut short.
Michael Gardyne hovered on the D of the penalty area as the Celtic defence cleared and smacked a volley against the base of Fraser Forster's left-hand post.
Goalkeeper Michael Fraser was quick off his line to block an effort by Ledley after the Welshman had been put through by Hooper.
Colin McMenamin was unable to get enough purchase on a shot that Forster claimed but the Highlanders' ability to get the ball forward quickly was causing Celtic problems.
However, Celtic doubled their lead seven minutes after the break.
Hooper forced a save from Fraser but County failed to clear their lines and Stokes's low cross was turned into the net by Boyd.
Pressure continued to be exerted on the Ross County backline and a late intervention by the experienced Grant Munro averted the danger posed by Forrest.
Celtic retained possession well in the closing stages and Fraser made an excellent save late on to deny substitute Paul George from close range.
Celtic manager Neil Lennon:
"I'm delighted, it wasn't easy. It never is coming to a First Division ground, the conditions were difficult to play in.
"We were very professional and it was exactly the reaction I was looking for from the players.
"The attitude and the manner we approached the game in was what I wanted.
"The football we played at times in the final third was very good. I'm also very pleased with the clean sheet.
"There were some very good individual performances."
Ross County manager Derek Adams:
"We started the game very well. We lost a bad goal but after that Michael Gardyne hit the post.
"In the second half, no complaints. I thought that Celtic thoroughly deserved to win the match.
"We went 2-0 down and there was only one winner after that.
"Celtic were far too strong for us. They're at a different level from us.
"I'm proud of the players and the way they stuck to their task. We caused them problems at times."