Bayern's Javi Martinez produced a dramatic 121st-minute equaliser to take the game to penalties and when Chelsea replacement Romelu Lukaku missed the 10th spot-kick after the previous successful nine, the Bavarians lifted the cup at the fourth attempt.
"One second changed everything, I am so happy for the players," said Bayern coach Pep Guardiola, refering to Martinez's equaliser.
Spain's Fernando Torres' early strike had been cancelled out by Franck Ribery's second-half thunderbolt while Chelsea's Ramires was dismissed in the 85th minute as it finished 1-1 after 90 minutes at Prague's Eden Arena.
"Of course I am very happy, especially after getting the award in Monaco, I am so happy for the entire team, it was a very special occasion," said man-of-the-match Ribery, who was voted European player of the year on Thursday.
The Frenchman was one of the survivors from the Bayern side who lost the 2012 Champions League final to Chelsea on penalties at Munich's Allianz Arena.
"The best team lost, but that is football," said Mourinho.
The Portuguese said he was annoyed about Ramires's dismissal for a second yellow card.
"There is a passion for football and if you are in love with the game, you don't kill a final by giving a red card like that," he said.
"For me, yes, (the player should not have been punished) because there were many other yellow cards during the match which the referee didn't give.