The Glaswegians were playing in their first UCL knockout tie in 12 years, and were looking to make up for lost time when Nicolas Kuhn rifled a left-footed strike past Manuel Neuer with a mere 25 seconds on the clock.
However, a rapturous Celtic Park was soon subdued when Adam Idah was penalised for interfering with Neuer’s vision as the ball flew past the Bayern shot-stopper.
Vincent Kompany’s men soon settled into their stride though, with Kasper Schmeichel twice called into action in quick succession to deny Michael Olise and Harry Kane.
It was hardly an onslaught, as the Hoops remained resolute in their defensive shape against a Die Roten side patiently probing in search of a way through.
Kane was involved in the thick of the action, as he headed wide from Olise’s corner, but it was the Frenchman who broke the deadlock in virtually the last kick of the first half.
The former Crystal Palace midfielder cut inside off the right flank and unleashed an unstoppable left-footed drive beyond Schmeichel, ending Celtic’s resistance just seconds before the half-time whistle.
Kane didn’t have to wait much longer to add his name to the scoresheet, doubling Bayern’s lead four minutes in the second half with a goal which Celtic’s defenders will not be keen on watching back.
The prolific marksman peeled away unmarked inside the six-box yard and converted a brilliant first-time finish past Schmeichel for his 29th goal across all competitions this season.
However, it looked like Celtic had a lifeline when referee Jesús Gil Manzano was called over to the pitchside monitor to view Dayot Upamecano’s challenge on Arne Engels in the box.
But much to Celtic’s frustration, Upamecano was deemed to have played the ball in a fair manner and no penalty was awarded.
Rodgers’ men pressed forward in search of a foothold back into this contest as Daizen Maeda took advantage of some defensive indecision and fired a shot right across the face of goal.
The Japan international did halve the deficit in the 80th minute, showing superb awareness to nod the ball home from point-blank range after Bayern failed to deal with an Engels corner.
Momentum looked to be swaying towards the Scottish champions heading into the closing stages, with Alistair Johnston forcing Neuer into a fine save at his near post.
But Kompany’s side held on to take a narrow lead into next Tuesday’s second leg and claim a fifth consecutive victory across all competitions. Meanwhile, this defeat ends a three-match winning streak for Celtic, who will now need one of the greatest results in the club’s history to progress into the round of 16.