After the France left-back put the visitors ahead with a superb second-half strike at Munich's Allianz Arena, the Bavarian giants hit back just 69 seconds later through a diving Mario Mandzukic header.
Thomas Muller then tapped Bayern into the lead before flying Dutchman Robben slotted the hosts third to keep the defending champion on course for the May 24 final in Lisbon.
"They scored the first goal so of course it was a tricky tie, but in the end 3-1 is a very good result," the former Chelsea winger told Sky Sports.
"I think it was very important we scored straight after they did. It was crucial today. It looked like a wake-up call because the first 10 minutes of the second half were a disaster the way we played.
"We were so slow, we were not there and you cannot do that in the Champions League - you will be punished. That's what they did, but it was a wake-up call and we scored three goals.
"We wanted to attack and play our football. They were there for the counter-attack - they are very dangerous with great players up front who are very quick.
"We had to be careful. In the first half we controlled the game, in the second half we were very, very bad and we got punished, but fortunately we turned it around."
Bayern coach Pep Guardiola said: "I knew it would be tough.
"It was a Champions League quarter-final and they had the goalie and eight players in the penalty area for most of the match. It happened in Manchester too.
"But we were patient and had some luck with our final touches.
"We decided to create a bit more passing in midfield and get the ball wide. They defended well, but when it got to 1-1 we played much better and we deserve
to be in the semi-finals.
"Now we are in the semi-finals, so am I very, very proud of my team."