With the scores locked at 0-0, the game went into extra-time at Schalke's Veltins Arena before Robben settled the tie to put his side in the final against Werder Bremen on May 15 as Bayern chase their 15th German Cup title.
"We were the better side tonight and we proved our fitness in extra time," said Robben, adding that his wife and newborn baby were the inspiration for his goal celebrations.
"I had the feeling things were going well for me and I was pleased with my goal, I dedicate the goal to my wife and my new daughter."
With Bayern set to face Manchester United in the first leg of the Champion League quarter-finals at Munich's Allianz Arena on Tuesday, this was a dress rehearsal for Louis van Gaal's side against their Bundesliga rivals.
The Royal Blues are second in the German league to leaders Bayern and battled their illustrious rivals for every ball.
Robben arguably had the best chance of the first 90 minutes when he was clear on goal with just ten minutes gone, but could not beat Schalke goalkeeper Manuel Neuer.
With less than 30 minutes left, French midfielder Franck Ribery came on for Bayern having shaken off an ankle injury, but could not break the deadlock.
Germany striker Miroslav Klose went close, but Robben finally settled the issue in the 112th minute.
The former Real Madrid star cut in from the right wing and beat the Schalke defence before drilling his shot past Neuer.
Holders Werder Bremen beat Augsburg 2-0 on Tuesday to seal their place in the final.