Robert Lewandowski scored twice and Kingsley Coman grabbed the other as Bayern claimed a record-extending 19th cup win to make up for the disappointment of losing last year's final to Eintracht Frankfurt.
Bayern coach Niko Kovac, who was in charge of Frankfurt at the time, boosted his chances of keeping his job in Munich after bringing a successful end to what had been a testing season.
Neuer, who missed Bayern's Bundesliga triumph the week before due to a calf injury, produced two stunning saves in either half when a goal for Leipzig might have been decisive.
Neuer somehow denied Yussuf Poulsen's header from close range in the 11th minute, when the ball crashed back off the crossbar, and got the better of Emil Forsberg in a one-on-one after the break, when the Swedish midfielder should have equalized.
"You couldn't write a better script," Neuer said.
Lewandowski opened the scoring with a header in the 29th, Coman doubled the lead in the 78th after Forsberg's miss, and Lewandowski wrapped it up with a chip over the goalkeeper in the 85th.
Arjen Robben and Franck Ribery went on late to make their last appearances for Bayern after 10 and 12 seasons respectively. Ribery became the only player to play in eight German Cup finals.
"They delivered a lot for the team and the club," Lewandowski said. "We have to be grateful."
Leipzig was appearing in its first, and had been hoping to celebrate the 10th anniversary of its founding date with a win. Thousands of fans made the relatively short journey from Saxony to cheer their side, though they found themselves outnumbered by Bayern fans at Berlin's Olympic Stadium, which hosts German soccer's showpiece every year.
"It's a huge disappointment," Leipzig chairman Oliver Mintzlaff said. "When you come here, you want to win."
The supporters had been warming up for hours before the encounter, downing beers and singing songs at various locations in the city before making their way to the game. Fans who arrived on chartered buses lit barbeques and shared crates of beer by the side of the road.
Leipzig coach Ralf Rangnick, the sporting director who took charge of the team on an interim basis before Julian Nagelsmann arrives from Hoffenheim, started with his strongest team, with American Tyler Adams lining up beside Kevin Kampl in midfield.
Leipzig looked sharper, fresher and quicker at the start, but one lapse was all Lewandowski needed to open the scoring. David Alaba whipped the ball in toward the penalty spot, where Lewandowski had too much space to head inside the left post.
Coman displayed a fantastic first touch to control the ball past the helpless Peter Gulacsi for Bayern's second, and Lewandowski ensured the result was emphatic when he claimed his competition-leading seventh goal. Lewandowski was booked for removing his shirt in the celebrations, but he accepted the yellow card with a smile.
"We worked hard for the whole season. In the end you could say it was a good season for us," the Polish striker said.
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