Goals from Lewandowski and Kingsley Coman sealed victory for Bayern in an enthralling game at the Olympic Stadium in Berlin.
Both Robben and Ribery came off the bench to make their final appearance in a Bayern shirt, and Lewandowski hailed the iconic duo after helping Bayern to victory with a goal either side of half time.
"These titles are for Franck and Arjen, they have done so much for the club and we have to thank them," the Polish striker told public broadcaster ARD.
The win sees Bayern lift the golden German Cup trophy for the 19th time in their history, just a week after securing a seventh successive Bundesliga title last Saturday.
"It was a difficult season at times, but I think we can now say that in the end, it was a very good season," said Lewandowski.
The campaign has been particularly tough for Bayern coach Niko Kovac, who has faced constant criticism both from inside and outside the club in his first year in charge.
Yet the fans chanted Kovac's name before and after the final whistle as Bayern ended their season on a high.
Both sides probed rather frantically for an opener in the first half, and Manuel Neuer denied Yussuf Poulsen with a reflex save on ten minutes.
It was Bayern who broke the deadlock, though, Lewandowski flicking in a superb header in the 29th minute after a fine move on the left from Coman and David Alaba.
Coman had a free run at goal and rounded goalkeeper Peter Gulacsi shortly before half-time, but his shot was cleared on the line by Ibrahima Konate.
Emil Forsberg had a similar chance for Leipzig just after the break, but came off second best in a one-on-one with Neuer.
A breathless second half saw chance after chance fall at either end, until Coman's 77th minute strike put Bayern out of reach.
The Frenchman found the top corner after elegantly bringing down a loose ball in the penalty area.
He later came off for departing veteran Franck Ribery, who was making his final appearance for Bayern alongside fellow winger Arjen Robben.
By that point it was already 3-0, as Lewandowski had broken free of the back line and lifted the ball over Gulacsi to secure victory five minutes from time.