The 2-0 win at Murray Park moves the Parkhead side to the top of the table.
Conor McManus and Joe Thomson both scored in a match that Celtic dominated throughout, with Paul McMullan also standing out for Stevie Frail and John Kennedy's young charges.
Celtic started well, with McMullan a constant danger, and the youngster hit the outside of the post early on.
It took them until after half-time to score, though, as a strong Joe Thomson shot clattered off the underside of the crossbar and fell to McManus, who scored the rebound.
McMullan was in the thick of it again shortly after, drawing a red card for Chris Hegarty after a poor challenge. Two minutes later, Celtic doubled their lead, as McMullan crossed from the right; it broke for Jamie Lindsay, whose shot was saved. But Thomson was there to stick in the rebound.
Gordon Durie, the Rangers manager, was bemused by his side's performance, a result which leaves them a point behind Hibs - who have played a game more - and two behind leaders Celtic. "The boys have been on a great run and I've just said to them that I personally didn't see that coming," he said.
"We were never at the races and we ultimately lost the game. We were fortunate to go in drawing 0-0 at half-time and I was delighted with that but we then deserved to lose it.
"We didn't get in about them and they dictated the pace of the game which I found disappointing. With their first goal, we let their winger get in too easily and when the shot came off the bar they reacted first.
"That was the story of our night and we were second-best most of the time. It's really frustrating that the majority of our players never turned up in this match and to lose the goals we lost was poor.
"Now we've got to dust ourselves down. We've been on a good run and we need to show a bit of character now because we've not had a result like this for a long time."