It was Dortmund's fifth straight win over Bayern, who were at full strength ahead of next Saturday's Champions League final against Chelsea, as they staked an even bigger claim to take over from the Bavarians as Germany's dominant domestic force.
Shinji Kagawa fired Dortmund into a third-minute lead after a double defensive blunder by Luiz Gustavo, before Dutchman Arjen Robben levelled with a penalty in front of more than 75,000 fans at Berlin's Olympic stadium.
Mats Hummels' penalty in the 41st minute restored Dortmund's lead and their top scorer Lewandowski struck either side of the break and again after Bayern winger Franck Ribery briefly cut the deficit with a sensational low drive in the 75th.
"We at Dortmund could not have imagined a better final," the club's beaming coach Jurgen Klopp told reporters.
Dortmund captain Sebastian Kehl lifted the trophy to mark his side's first German Cup triumph since 1989 as his team mates cheered wildly with German president Joachim Gauck sucked into the celebrations.
"We were ice cold in our execution, scored great goals. It's difficult to put into words what we're feeling at this moment," said Klopp after his first German Cup victory as a coach.
"We should not be complaining as our overall defensive behaviour was catastrophic," Bayern coach Jupp Heynckes told reporters. "We have to accept that we did not deserve to win. But now we have a different target in our sights and until the Champions League final we will recover from the shock."
"That will be a completely different game," said Heynckes.
