Kompany hails composed Bayern's Champions League win at Celtic
Bayern boss Kompany was concerned about the potential for Celtic to be fuelled by the raucous 60,000 sell-out crowd in Glasgow.
But the Bundesliga leaders produced a mature display that took the sting out of the Celtic roar for long periods.
Michael Olise's thunderous drive into the roof of the net from 15 yards put Bayern ahead on the stroke of half-time.
England captain Harry Kane doubled Bayern's lead with a close-range volley from Joshua Kimmich's corner four minutes after the interval.
At that point Bayern were in complete command, but they eased off in the closing stages and Daizen Maeda's 79th minute header gave Celtic a glimmer of hope.
Bayern keeper Manuel Neuer had to make a good save from Alistair Johnston to ensure the visitors held onto their lead in a frantic finale.
Kompany was relieved to hear the final whistle, but insisted he was delighted with the way Bayern inflicted Celtic's first home defeat in 33 games in all competitions.
"It's cliche to say, but it's the first half of the game. Overall, it's a really great result," he said.
"I know this place, I know it transcends performance sometimes. We dealt really well with the atmosphere for long spells before it could go too 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.
"The fans haven't seen many losses here, especially in Europe, so we appreciate the importance of the game and the victory."
'One more shot'
Bayern will be favourites to advance to the last 16 when Celtic visit Munich for the second leg on February 18.
But before then, Kompany's men face a crucial clash on Saturday with champions Bayer Leverkusen, who are eight points behind Bayern in the title race.
"We're also pretty decent at home so we'll look to play on that as well," Kompany said.
"We have an important game on Saturday, so we'll switch our focus to that."
Celtic manager Brendan Rodgers took heart from the way his side recovered from their ultra-cautious start.
Rodgers is confident the Scottish Premiership leaders can rattle Bayern in the return as they look to reach the last 16 for the first time since 2012-13.
"At 2-1, Bayern Munich are hanging on. That's what we can take from it. We've got one more shot," Rodgers said.
"It's a new level for a lot of the players, they've done absolutely great. Let's go and have no regrets and see where it takes us.
"By the end, Bayern are happy for the final whistle. We've shown we can play football and be a threat. We're still right in the tie."
Celtic treated Bayern with too much respect in the first half, sitting deep for long spells rather than taking the fight to their star-studded opponents.
Rodgers admitted it was only when Celtic went two goals behind that they finally threw caution to the wind.
"The last 25 minutes, we started to press the game and you see what it brought -- aggression. Our football starts to come alive and then we get the goal," he said.
"With Bayern's quality and the number of goals they can score, sometimes these can be out of sight but that wasn't the case at all.
"We were always going to be the underdog. But what must never happen is, you must never be broken. We kept fighting and got a reward for that."