Vincent Kompany's side dominated the first half and a stunning strike from Olise gave them a deserved lead before Kane maintained his red-hot scoring streak after the interval.

Daizen Maeda netted in the closing stages to give Scottish champions Celtic hope of a miraculous escape act in the second leg.

The Japan forward was only playing after his two-game ban was unexpectedly reduced on appeal this week following his red card against Young Boys.

But Bundesliga leaders Bayern will still be favourites to advance to the last 16 when Celtic visit Munich on February 18.

Celtic finished just three points behind Bayern in the league phase, yet there is a huge gulf between the clubs' European pedigree.

Six-time European champions Bayern lead the Bundesliga by eight points, while Celtic are attempting to reach the last 16 for the first time since 2012-13.

For the first 70 minutes that chasm in quality was clear to see before Bayern eased up in the closing stages, nearly allowing Celtic back into the tie as a result.

Celtic's first defeat in 33 home matches across all competitions was a welcome confidence boost for Bayern, who had shown a weakness on the road in league phase defeats at Barcelona, Aston Villa and Feyenoord.

Celtic thought they had preyed on that weakness when former Bayern forward Nicolas Kuhn fired home from 20 yards inside the first minute, only for the goal to be disallowed for offside against Adam Idah.

Bayern could have been swept away by the tidal wave of noise cascading down from the Parkhead stands.

But Kompany's men are too experienced for that and they quickly seized control with a steady diet of possession.

Predatory Kane

Olise nearly put Bayern ahead with a powerful strike that forced Kasper Schmeichel to save at his near post.

Schmeichel was forced into action again moments later as Kane's curler was pushed away by the Dane.

Bayern were camped on the edge of the Celtic area as they probed for an opener.

Kane should have broken the deadlock when he nodded Olise's corner into the side-netting from close-range.

But the England captain's blushes were spared by Olise on the stroke of half-time.

Cutting inside Greg Taylor to reach Dayot Upamecano's long pass, Olise unleashed a blistering drive that arrowed into the top corner from 15 yards.

It was no more than Bayern deserved for their territorial dominance and the Germans struck again four minutes after the interval.

Joshua Kimmich's corner swung all the way to the far post without a Celtic clearance and the unmarked Kane guided a composed volley past Schmeichel from five yards.

Kane's typically predatory finish took his season tally to 29 goals in 29 games in all competitions.

Upamecano's challenge on Arne Engels prompted a long VAR check for a potential penalty, but Celtic's appeal was correctly rejected.

That was the signal for Celtic to mount a late bid to haul themselves back into the tie.

Bayern had grown increasingly sloppy and Rodgers' men earned their lifeline in the 79th minute when Maeda met Yang Hyun-Jun's shot with a header from virtually on the line.

Kompany's men had to cling on for dear life in the final minutes, with Manuel Neuer producing a crucial save to deny Alistair Johnston.