The German side won 3-1 on penalties after a 2-1 win to Madrid on the night at the Santiago Bernabeu saw the game end 3-3 on aggregate.
"I am very proud of my team, because when you start a game like this and after 14 minutes you are 2-0 down, then you have to be mentally strong to cope with it and recover from it," Robben told Sky Sports News.
"I think, overall, the rest of the game, we played some really good football.
"In the end, I think we can say it is deserved. We were really hungry. Everybody was really aggressive."
Munich will have the advantage of playing in their own Allianz Arena when they face Chelsea in the final on Saturday, May 19.