Rooney slotted home the only goal of a game, which United dominated, clinically firing home 16 minutes from time.
Nemanja Vidic had been sent off four minutes into the second half for an elbow on Claudio Bieler.
The goalscorer told Channel Five: "It's fantastic. It's a great achievement by us. We are really delighted and proud that we are the world champions."
Rooney admitted it was a tough game, especially after Vidic's sending off.
But he added: "To score the winning goal is a great feeling for myself."
Reds captain Rio Ferdinand said: "Now we can say we are the best team in the world.
"We knew they could create problems but I think today we kept them down to a minimum of chances."