The German league leaders saw their 2-1 first leg lead wiped out inside the opening 10 minutes at Old Trafford courtesy of strikes from Darron Gibson and Nani, who later added a second to give the Red Devils a 4-2 aggregate advantage.
However, Ivica Olic pulled a goal back before half time against the run of play.
And after United's right-back Rafael had been dismissed shortly after the interval for a second bookable offence, Louis van Gaal's side proceeded to dominate and eventually grabbed the decisive goal that took them through to the next stage with Arjen Robben spectacularly volleying home.
NEWS:Robben stunner sends Bayern through
"I was very pleased to see that second goal go in, but at the time, I think we deserved it," Van Bommel said.
"In the first half, we were not so good, we were very lucky that we scored a goal just beore half time. That was really important and after the red card we played very well. It isn't so easy against 10 men.
"We weren't surprised at United's start and we knew it would come. But we didn't play well first half and like I said before, we had to be lucky."
The Dutchman was yellow-carded during the encounter and is now suspended for Bayern's semi-final first leg against Lyon - a decision he felt was harsh.
"In my opinion it wasn't a yellow card. It was a mistake of the referee. But OK, I have to accept it and we go on."
