The visitors opened the scoring in the 23rd minute through Christoph Baumgartner, but Kuhn curled home a fine equaliser for Celtic before netting again before half-time.

Reo Hatate gave Celtic some breathing space as Brendan Rodgers' side moved onto seven points after four games, boosting their hopes of reaching the knock-out phase for the first time since 2013.

"If you told us after the Dortmund game we'd have four points from the next two games, I don't think too many people would have said we would have done that," said Celtic manager Rodgers, whose side were thrashed 7-1 by Borussia Dortmund in their second match.

"It's a great next step and gives us great encouragement to go and make the play-off stage, but there is still so much to do."

Leipzig, who suffered their first Bundesliga defeat of the season at Borussia Dortmund at the weekend, are one of six teams still yet to get off the mark in the Champions League this season.

"Something was missing, you have to be honest about that," Baumgartner told DAZN.

"Everyone has to look in the mirror and ask themselves if that was enough. Because we were too sloppy, too inaccurate in our passing game and made too many easy mistakes."

Leipzig struck first for the third time in the competition as Baumgartner stooped to nod home after Kevin Kampl's corner was inadvertently flicked on by Celtic defender Cameron Carter-Vickers.

Willi Orban almost doubled Leipzig's lead shortly afterwards, heading another dangerous set-piece narrowly wide.

But Celtic hit back 10 minutes before half-time as Kuhn jinked inside and bent a wonderful strike in off the far post.

Kuhn almost turned creator a few minutes later, playing a magnificent through ball with the outside of his right boot to Daizen Maeda, but the Japanese forward could only blast over with just Leipzig goalkeeper Peter Gulacsi to beat.

The German completed the first-half turnaround himself in the first minute of added time, firing home Greg Taylor's cutback after Maeda completely missed the ball with his attempted shot.

Both sides went close early in the second half, with home 'keeper Kasper Schmeichel denying Benjamin Sesko an equaliser and Gulacsi keeping out Hatate's low strike.

But Hatate did give Celtic a two-goal cushion with 18 minutes remaining, turning the ball home after Gulacsi inexplicably spilled Alistair Johnston's tame cross.

The Scottish champions passed up further opportunities to score late on, but were able to celebrate a crucial victory.