The German international scored the winning penalty at the Santiago Bernabeu on Wednesday night following a gruelling 120 minutes against the Spanish giants.
"We're all exhausted, but overjoyed," Schweinsteiger told the club's official website.
"I'm speechless, I'm overwhelmed. It was a dream which has become reality."
Bayern will now take on Chelsea at the Bundesliga side's own Allianz Arena for European club football's top prize.