"It was a fantastic performance and I can only say congratulations to my players and hope the Real fans are happy with what they saw," said a delighted Mourinho.
"We have nine points but of course we want to be first in the group and to achieve that we have to continue winning matches."
With 16 European Cup trophies between them it was a genuine clash between two heavyweights of the game as the competition's two most successful teams went head-to-head.
But nine-time champions Real were the dominant force throughout at the Santiago Bernabeu as they produced an accomplished display, with Cristiano Ronaldo at the creative heart of their attacking excellence.
They already look a more disciplined side under Mourinho but they have lost none of their menace in the final third of the pitch as they threatened to carve Milan open almost at will.
After the wasteful Di Maria fired over, two goals in two minutes just before the quarter-hour mark earned Real the advantage their dominance deserved and gave Milan a mountain to climb.
Firstly Alexandre Pato conceded a needless free-kick on the edge of his own box and as the Milan wall parted inexplicably, Ronaldo drove the ball mercilessly through the gap and into the net.
Sixty-five seconds later it was 2-0 as Ronaldo scampered down the left and crossed for Ozil, who saw his volley into the floor loop in via the shoulder of the unfortunate Bonera.
Ozil celebrates scoring his first European goal in a real shirt
Higuain headed over as Real poured all over the visitors, before Milan gave an all-too-brief exhibition of the class they undoubtedly possess.
Pirlo's brilliantly dipping free-kick brought a superb save from Iker Casillas, who touched it on to the bar, before Ronaldinho - otherwise desperately disappointing - flicked Seedorf into space inside the Real box only for the Dutchman to balloon haplessly over.
That apart it was all Real as Ronaldo continued to cause mayhem, with one electric break ending in a sizzling left-foot shot from outside the box that Marco Amelia had to push behind at full stretch.
Amelia, in for the injured Christian Abbiati, was being given a thorough workout on his Milan debut and he saved well twice from Di Maria, while Higuain planted a presentable header into his grateful arms from six yards.
Milan hauled off Ronaldinho with 20 minutes left and sent on ex-Real forward Robinho, but save for one shot that Casillas adeptly turned behind the Brazilian could not haul his team back into it and his old employers are now in pole position in the group.
Milan coach Massimiliano Allegri admitted they had been second best: "We had goalscoring chances but you have to do more than we did to hurt Real Madrid.
"The first quarter of an hour was crucial. They scored two goals and we didn't know what to do."
Real Madrid: Casillas, Arbeloa, Pepe, Carvalho, Marcelo, Alonso, Khedira, Ozil (Lassana Diarra 82), Di Maria (Granero 87), Ronaldo, Higuain (Benzema 89). Subs Not Used: Dudek, Mahamadou Diarra, Albiol, Pedro Leon.
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