Kilmarnock slipped to their first defeat in over a month as they let a lead a Celtic Park slip.
Eamonn Brophy fired Angelo Alessio's men ahead before an Odsonne Edouard double and Ryan Christie strike ensured they were heading back to Ayrshire empty-handed.
The hosts had a great chance to take the lead in the third minute when Ryan Christie fired a shot from 20 yards out just past the post.
Killie's on-loan QPR left-back Niko Hamalainen had a decent effort blocked out for a corner in the eighth minute, the resulting set-piece was comfortably cleared.
In the 15th minute James Forrest smashed the post with a low drive towards Laurentiu Branescu's bottom right-hand corner.
Forrest had another chance three minutes later with another low shot, but Romanian stopper Branescu got down well to parry it away.
Christie then pulled another good save out of the on-loan Juventus goalkeeper with a powerful effort from 25 yards.
In the 27th minute Christie was booked for diving as he went down after a challenge from Stephen O'Donnell.
Just after the half-hour mark Killie took the lead thanks to a wonderful Brophy finish.
Christie whipped in a very inviting cross from a free-kick in the 40th minute, but a handful of Celtic players attacking the ball failed to knock it home.
In the 44th minute Killie were dealt a devastating blow when Edouard headed home well.
Seven minutes into the second-half the hosts were fired ahead when Edouard slipped the ball past Branescu from a tight angle to double his money.
Four minutes later Christie put the Scottish Premiership Champions on easy street as the ball fell to him for an easy tap in after a stunning Branescu save.
In the 69th minute Brophy had a great chance after cutting inside, but Fraser Forster did well to tip it over for a corner.
Osman Sow was brought down in the box in the 85th minute, but Alan Power saw his penalty saved.
