The home side went toe-to-toe with the in-form Bundesliga outfit but after a mistake by midfielder David Turnbull in the 25th minute, 19-year-old Ecuador defender Piero Hincapie gave the visitors the crucial lead.
Then 18-year-old star midfielder Florian Wirtz added a second for Gerardo Seoane's men in the 36th minute and despite strenuous efforts by Celtic to get back into the match – and they had several chances throughout – Lucas Alario added a third from the spot on the hour mark before substitute Amine Adil thrashed in a breakaway fourth in added time.
Ange Postecoglou's side have now won only two of their last eight games in all competitions, including a 4-3 defeat by Real Betis in their in their Group G match.
Celtic are in sixth place in the cinch Premiership, six points behind leaders Rangers and return to league duty at Aberdeen on Sunday and will be boosted by aspects of their performance and by having some key players back.
There was a lot of speculation around which of the Celtic players recovering from injury would start the game.
Hoops fans in a packed and noisy Parkhead were delighted to see the return of captain Callum McGregor and striker Kyogo Furushashi with Greek striker Giorgos Giakoumakis, yet to make his debut, on the bench.
Former Celtic right-back Jeremie Frimpong was in the Bundesliga side but striker Patrik Schick, who scored a double – including a wonder goal – for Czech Republic at Hampden Park in their opening Euro 2020 game, was among the substitutes.
The game burst into action. Within a couple of minutes Leverkusen striker Alario had the ball was in the back of the Celtic net after keeper Joe Hart had spilled a shot from Wirtz, only to be ruled offside.
Seconds later at the other end, Furuhashi sped into the box and rounded keeper Lukas Hradecky and set himself to stroke the ball into the empty goal but was robbed at the last second by defender Jonathan Tah before Hradecky made saves from Furuhashi and defender Carl Starfelt.
Hart had to be smart a close-range effort from Moussa Diaby but he had little chance when the German side took the lead.
Turnbull slipped near the byline which allowed Mitchell Bakker to cut the ball back for Hincapie to sweep the ball in from 16 yards despite the efforts of young left-back Adam Montgomery, also back in the side.
The Celtic defence was in disarray when Paulinho played in Wirtz who knocked the ball past Hart and into the net which left a very different complexion on matters.
Furuhashi tested Hradecky again at the start of the second half with a shot on the turn before the keeper made an even better save from Jota with winger Liel Abada missing the target with the follow-up shot.
A penalty shout when Frimpong knocked over Turnbull was ignored by referee Marco Di Bello but when the former Hoops defender drew a save from Hart with a drive, Paulinho's shot from the rebound was judged to have hit the arm of defender Cameron Carter-Vickers.
Alario sent Hart the wrong way with the spot kick and the tie was effectively over.
Moments later, Hradecky brilliantly saved a header from Jota and Furuhashi hit the side netting with the rebound and another great chance was missed.
In the 74th minute, amid a glut of substitutions, Giakoumakis replaced Turnbull but it was Hart who had to make the next save, a header from Schick, who had also come on before Adil hammered in a fourth on the break. Now Celtic must regroup and get ready for the Dons.