Arjen Robben scored the winner in the 89th minute, finishing off a game that had seen Mario Mandzukic put the Bavarians into the lead before Ilkay Gundogan's equaliser from the penalty spot.
"What we've achieved so far this season – because it's not over yet, we still have the German Cup final next weekend – has been outstanding," he told reporters.
"There's never been a Bundesliga team that's consistently played at such a high level, winning the championship by 25 points, breaking almost all records.
"Today my team were determined to win; look at the generation of Philipp Lahm, Bastian Schweinsteiger – the time had come to win something like this. They've won a lot of caps and domestic trophies, but this was the first international competition. I'm really pleased for my team."
Bayern can complete the treble when they take on Stuttgart in the German Cup final on June 1.