Although Bayern dominated possession and had more shots on target in the match, they struggled to find a way past Joe Hart until the closing minutes.
It took a deflected strike from former City defender Jerome Boateng to seal the victory for the Bundesliga champions and Lahm was delighted their patience was rewarded.
The 30-year-old also praised the effort shown by their opponents and admitted Bayern needed a bit of luck to secure a crucial win over their Group E rivals.
"It was a bit difficult to start with because Manchester saw a lot of the ball and we were chasing shadows," he told the club's official website. "But we switched things around and it was a lot better.
"We had some huge chances in the first half, and we had the better chances over the full 90 minutes.
"We've played one of the best teams around and they're very dangerous. You don't win matches like that just by turning up.
"We worked hard and we're pleased with the win. We wanted to open with a win, although it's obviously lucky when the winning goal comes so late."
Bayern will look to make it two wins out of two when they face CSKA Moscow in the second round of fixtures on September 30, the Russian side aiming to bounce back from their 5-1 mauling by Roma on Wednesday.
