Kingsley Coman scored the decisive goal after 59 minutes of play to ensure Bayern would emulate their treble-winning season of 2012/13.
Both sides have chances in the first-half but were unable to convert.
Up stepped Coman against his former side, with the Frenchman connecting on Joshua Kimmich's cross with a back-post header.
Thomas Tuchel's side will be kicked themselves after possessing more than enough chances to win the game, and potentially the club's first Champions League crown.
Speaking after the game, Bayern Munich star Serge Gnabry told BT Sport: "Winning the trophy today is the best thing that could happen to us. We worked incredibly hard and in the end it came through to be the best team in Europe.
"It's the final, Paris came out wanting to win and so did we. Nobody is going to give up easily but we came through.
"As long as it was 0-0, it was always going to be open for us. Of course, they were going to have chances. It was lucky they didn't score and lucky we did score."
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