Bayern will meet Werder Bremen, who beat second division Augsburg 2-0, in the final in Berlin on May 15.
Dutchman Robben grabbed the winner in the 112th minute of an otherwise lacklustre game when he raced some 40 metres down the right, cut back into the middle and released a curling left-foot shot into the top corner.
"Robben dribbled a lot and still had power to decide the match like that. That's superb," Bayern coach Louis van Gaal told reporters.
"I think we were superior in both tactics and fitness. We were patient and luckily he (Robben) made that goal," he told ARD Television.
Both teams laboured with their quick passing game on an extremely rough pitch and Schalke's solid defence, the best in the league this season, left Robben and Frenchman Franck Ribery, who came on in the second half, with little space to move.
"I felt good and when you get the ball and have that space you have to take it," Robben said. "I think we were the better team ... and in extra time we were as fit as Schalke."
