Rooney disappointed to concede second goal
United, chasing a place in their third successive Champions League final, fell behind early in the round-of-16, first-leg match at the San Siro before a freak goal by Paul Scholes put them back on level terms.
Rooney's second-half double silenced the home crowd before a classy goal by Clarence Seedorf rekindled Milan's slim hopes of overhauling the deficit at Old Trafford in three weeks' time.
"We were disappointed to concede a goal towards the end but in the end it's still a great result."
United were fortunate not to fall further behind in the first half after Ronaldinho's opener but tactical adjustments paid off as the English champions gradually took control of the game.
"We showed them too much respect in the first half and sat back a bit too much, we didn't take the game to them or push forward enough and that was the most disappointing thing," Rooney said.
"We didn't start too well but then we got back in the game and in the second half we controlled them so we deserved to win. "We are proud to be the first United team to win (at Milan) and I am delighted to have scored two goals."
The brace took Rooney's tally for the season to 25 goals in all competitions and when asked if he thought he could win the 2010 FIFA World Player of the Year award, he said: "It would be nice but I am just aiming to be unselfish and help the team first. If the award comes with it I'll be delighted."
United, also aiming for a record fourth successive Premier League title, are one point behind leaders Chelsea and play Aston Villa in the League Cup final on February 28.