The Germany international confessed that the Bavarians had nothing left after giving their all in securing a place in May's tournament decider at the Allianz Arena
The Bavarians appeared to be heading for an exit after falling 2-0 down on the night against Los Blancos, but Arjen Robben levelled the tie with a penalty before Schweinsteiger held his nerve to decide the subsequent shoot-out in the visitors' favour and earn Bayern a place in next month's final in their own Allianz Arena.
"We're all exhausted, but overjoyed," the Germany international told his club's official website.
"Ever since it was announced that the final would be in Munich, we all had only one target: to get there! Now we have one more match at home, and we want to win it."
Meanwhile, Schweinsteiger's team-mate, Mario Gomez, was almost lost for words after overcoming Madrid in their own backyard.
"I'm speechless, I'm overwhelmed," the striker confessed. "It was a dream which has become reality.
"In this atmosphere and in this situation, there’s nothing better than getting through in a penalty shoot-out."