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.
1. Joe Hart - 5
Rarely tested in the first half. Beaten by a Ryan Kent effort which hit the base of the post, while there was one errant pass. Question marks over the opening goal. Got a hand to Filip Helander's header but not strong enough to keep it out. Photo: Rob Casey - SNS Group
2. Anthony Ralston - 7
The rebirth of the right-back continues. Not as refined on the ball as his team-mates and passed up a couple of good crossing opportunities but defensively flawless. So strong in one v one against both Kent and Borna Barisic. Photo: Craig Foy - SNS Group
3. Stephen Welsh - 6
Didn't do much wrong. Produced a brilliant interception in the first half to prevent Rangers breaking in behind. Photo: Rob Casey - SNS Group
4. Carl Starfelt - 5
Guilty of being drawn into physical battles he wasn't going to win and ended up giving up cheap fouls. Beaten in the air by Helander for the Rangers goal. Photo: Alan Harvey - SNS Group