The Spaniards, trailing 2-1 from the first leg, raced into to 2-0 lead inside 15 minutes thanks to a double from Cristiano Ronaldo, but Bayern levelled the tie through Arjen Robben's penalty just before the half hour mark.
Neither side could then break the deadlock as the game drifted into extra-time and finally penalties and after Ronaldo, Kaka and Sergio Ramos had all missed for the hosts, Bastian Schweinsteiger rattled home his spot-kick which means Bayern will become the first side to host a Champions League final on their own ground when they face Chelsea on May 19.
And Mourinho believes the extra strain put on his players from beating Barca on Saturday, whilst Bayern rested a host of players in their 2-1 win over Werder Bremen, proved decisive.
He said, "We deserved to win, mainly because the matches were very even and it is a completely different situation when a team plays for a championship against a side that doesn't.
"I had the same situation when Chelsea played Liverpool in a semi-final. Liverpool were 30 points off the top and played a reserve team at Fulham whilst Chelsea fought every game to win the championship that season.
"If we weren't playing to win the title, we would have had fresher players, but the team were fantastic in terms of their physical condition.
"Barcelona played the Clasico and then two days later lost to Chelsea, this has incredible value.
"You only have to look at the final which will be played between the fifth placed team in England against the second placed team in Germany, both of which used second choice players at the weekend."
However, Mourinho praised his former club for their stout defensive performance against Barca at the Nou Camp on Tuesday and was scathing of those who criticised their tactics.
"John Terry will be missing but Chelsea are there and that is the most important thing. Chelsea were heroes yesterday.
"People who criticise them know nothing about character, effort and how a team with 10 men can resist physically, mentally and tactically. It reminded me of my own Inter Milan team two years ago.
"The Chelsea fans are fantastic, as are the German fans so I hope they enjoy it and it is a great final, but I hope Chelsea win."
Mourinho also hinted that the disappointment of falling at the semi-final stage for a second consecutive season may encourage him to stay in Madrid.
"I want to win this title with this team. Reaching two semi-finals is not bad but of course we want more and my feelings are that the group still has room for growth.
"I will discuss my ideas with the club of how to move us forward on and off the pitch and if the club think I can contribute something, which I get the feeling they do, then I think I will continue."
Bayern boss Jupp Heynckes was delighted with how his players reacted to falling two goals behind so early on.
He said: "In the first 15 minutes Real were ahead of us. We were 2-0 behind and everyone knows how difficult it is to play in the Bernabeu.
"After this though we dominated, we managed to release the pressure and we improved as the game went on.
"We improved in defence and had opportunities. In 120 minutes we deserved to reach the final."
Heynckes called on UEFA to change the rule which sees players miss the final if they pick up their second yellow card of the competition in the second leg of the semi-final.
David Alaba, Luis Gustavo and Holger Badstuber will all miss the final for the Bavarians after being booked and with Chelsea also missing four key players, the German coach believes it is time for the governing body to take action.
He explained: "That is a problem with the rules, I feel the rules should be changed.
"It is a shame the final can't be played by the best players both for Chelsea and Bayern.
"The yellows shouldn't be carried forward to the final, these three players have been exceptional for us."