The Hoops were hoping to forget their chastening 7-1 Champions League defeat at Borussia Dortmund in midweek.
But Brendan Rodgers' men had to come from behind in the Highlands after conceding for the first time in seven league games.
Ronan Hale needed two goes to score from the penalty spot to put County in front after a Liam Scales handball.
The striker's first spot-kick was saved by Kasper Schmeichel, but the Danish international had strayed off his line in making the stop.
Hale made the most of the reprieve to fire the home side in front just before half-time.
Rodgers had made just three changes from the side that were taught a lesson by last season's Champions League finalists.
And it was not until he injected fresh legs from the bench that Celtic came to life in the final half hour.
Alistair Johnston deflected in Callum McGregor's shot to level 15 minutes from time.
The same two players then combined to free Kuhn down the right and the German winger cut inside to score his sixth goal of the season.
"We looked like we were looking for an easy game, and that's not what this team is about," said Rodgers.
"In the second half, especially with the introduction of the new players coming in, we got real good energy in the game, started to run more, started moving more and then our quality paid off in the end."
Next up for Celtic is a top-of-the-table clash with Aberdeen, who also made it seven wins from seven games in the league by beating 10-man Hearts 3-2.
Aberdeen got off to the perfect start in the second minute through Topi Keskinen, but managerless Hearts hit back through Frankie Kent.
Hearts sit bottom of the table, without a league win this season, but went in front when Blair Spittal rifled in a low shot from the edge of the box.
This time the Dons quickly fought back through Nicky Devlin.
Jorge Grant was sent off 15 minutes from time and Aberdeen took advantage when Ante Palaversa fired home the winner in the 88th minute.
"The whole of Pittodrie was bouncing," said Aberdeen boss Jimmy Thelin, who has won his first 13 games in charge in all competitions.
"I think everybody could feel the atmosphere today was something extra, and it was amazing."
Rangers remain five points adrift of the leaders after Vaclav Cerny struck twice to beat St. Johnstone 2-0 at Ibrox.
