Malky Mackay's men frustrated the hosts for over an hour before Cameron Carter-Vickers’ deflected effort opened the scoring, then an Albian Ajeti double sealed their fate.
County had contained Celtic well prior to that, and had carried a threat of their own, with sub Dominic Samuel denied an equaliser when Joe Hart saved his header.
“It was one of those frustrating games where you have to put away chances like that," Samuel said.
“As a striker, you always want to be putting those situations into the back of the net.
“I thought I would just get a connection on it and use the wet surface to let the ball skid into the goal.
“He’s a very good, experienced goalkeeper and he got down early. There was not much more I could have done other than just let it hit the target."
Samuel joined from Gillingham in the summer and is one of 12 new signings bedding in at County.
“It’s a fairly new team," he added. "A few players are still getting to gel with each other. I feel like once the games keep on coming we will get better.
“If we are creating these kinds of chances against the bigger teams, once we start playing other teams our confidence will just shoot up.
“On the attacking side of things we have been doing quite well, but we have to start putting things into action and start picking up the points."