Vincent Kompany celebrated a "great result" as Bayern Munich survived a tough test at Celtic, though the visiting boss warned it is only half-time in their Champions League tie.

Michael Olise and Harry Kane were on target either side of half-time at Celtic Park before Daizen Maeda offered the hosts hope when pulling a goal back 11 minutes from time.

Bayern will take a slender 2-1 lead into Tuesday's return meeting of their play-off tie, and Kompany was satisfied to escape Celtic's fortress with a hard-fought victory.

"It's cliche to say, but it's the first half of the game. Overall, it's a really great result," Kompany said at his post-match press conference.

"The fans haven't seen many losses here, especially in Europe, so we appreciate the importance of the game and the victory."

Kompany always expected a difficult challenge, acknowledging how important the home crowd can be when inspiring Celtic past their opponents.

"I know this place, I know it transcends performance sometimes," he added. "We dealt really well with it for long spells of the game, didn't let the game go crazy.

"The last 10 minutes we had to defend and we did that well. Of course, you can feel in those moments, if the game comes alive it is a special place."

Manuel Neuer was still required to deny Alistair Johnson in the dying seconds, though Bayern will have felt they deserved the win.

However, the home side accumulated 1.34 expected goals (xG) to Bayern's 0.94, suggesting Brendan Rodgers' side shaded the contest on the quality of chances they created.

Kompany seemed to have no fears, however, knowing Celtic must make the trip to Munich next week.

"We're also pretty decent at home so we'll look to play on that as well," he concluded.

"We have an important game on Saturday, so we'll switch our focus to that."

Rodgers, meanwhile, was disappointed with how Celtic defended for Kane's goal, with the striker converting after Joshua Kimmich's corner delivery.

"The corner [for Harry Kane's goal] was disappointing," he told his post-match press conference. "One of the world's best strikers shouldn't be left free in the box, that's for sure."