Bundesliga top scorer Lewandowski headed the champions into the lead in the 29th minute, wrong-footing keeper Peter Gulacsi, and Kingsley Coman's superb first touch and shot in the 78th made it 2-0.
Lewandowski sealed the win with a fine run and superb finish in the 85th minute.
It was the 12th time Bayern won the double as coach Niko Kovac, who lifted the Cup with Eintracht Frankfurt last year and led them to the 2017 final, became the first man to win the double with Bayern both as player and coach.
But Bayern also had to thank keeper Manuel Neuer who returned from injury and made two superb point-blank saves to protect their one-goal lead.
"It was the toughest opponent we could have had and we saw that in the opening 30 minutes," said Kovac, whose future at the club looks more secure after winning two titles in the past week.
"Manuel Neuer kept us in the game with two superb saves but overall the victory was deserved," he said. "We had tough times back in October and November so compliments to the team. To be here with two titles makes me very proud."
Leipzig enjoyed the better start and had the first big chance with Yussuf Poulsen's header palmed on to the crossbar by Neuer in a reflex save before Lewandowski put them ahead.
Coman came close to a second goal but his effort on an empty goal was headed away at the last moment by Ibrahima Konate.
In a highly entertaining game, Neuer again come to the rescue by stopping Emil Forsberg in a one-on-one situation.
The Bavarians hit back with Coman to create a two-goal cushion before Poland striker Lewandowski struck again to kill off any lingering Leipzig hopes.
"I am very satisfied with the way we played well and won clearly," Lewandowski said. "We had the game under control and the second half we pushed a bit more and won the Cup.
"We gave everything for the double. At the end we can say that this season was very good for us," Lewandowski said.
Bayern winger Franck Ribery, who is leaving Munich along with Arjen Robben, came on as a late substitute to make a record eighth German Cup final appearance in his last game for the club.
(Reporting by Karolos Grohmann, editing by Ed Osmond)
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