Filip Helander’s goal was the difference, the Swede rising highest to meet Borna Barisic's corner in the second half.
Celtic started well but lost their control after the interval and struggled to test Robby McCrorie until they fell behind.
There were a number of poor performers for the visitors and Ange Postecoglou's men have started the season with the fewest number of points since the 90s. They find themselves sixth with six points after four games going into the international break.
How did the 15 players involved rate out of ten?
Player ratings scale - 10: One of the best performances you've seen. 9: Outstanding. 8: Very good. 7: Good. 6: Average. 5: Poor, 4: Very poor. 3: A terrible day at the office. 2: Abysmal. 1: One of the worst performances you've seen.
9. Kyogo Furuhashi - 6
Always the sense of something happening when he got on the ball. Created a great chance for Edouard and was the team's most dangerous player. Passed up good opportunities, however, and seemed reluctant to go at Leon Balogun when playing left. Photo: Rob Casey - SNS Group
10. Odsonne Edouard - 4
In what could be the Frenchman’s last game for the club, he’ll likely feel he could have done a lot better. A poor touch early on and then missed a sitter, the ball hitting clumsily off his heel. Anonymous after the interval. Photo: Alan Harvey - SNS Group
11. Liel Abada - 4
Played on the periphery of the game to the extent that when he was replaced he probably reminded quite a few fans he was actually on the pitch. Photo: Rob Casey - SNS Group
12. Tom Rogic - 6
Difficult for the Australian to come off the bench and make an impact with the game played at such a pace. Photo: Craig Foy - SNS Group