Argentine striker Diego Milito scored in each half as Milan made it a triple trophy season adding the Champions League to the Serie A title and the Coppa Italia.
Milito felt Inter deserved to overcome their German opponents and also dedicated the triumph to the club's president Massimo Moratti.
"It's an incredible happiness," Milito told Sky Italia immediately after the game.
"I'm happy for Inter. We aimed a lot for this cup. The president deserved it, too.
"I'm absolutely happy. I've always tried to struggle and to give my best.
"We deserve it, it's deserved by everyone."
However, Milito was unable to give any guarantees about his long-term future as he prepares to head off to represent Argentina at the World Cup.
"In football, anything is possible. We will see next season."