Liverpool, and in particular Raheem Sterling, missed a host of good chances with David De Gea in fine form.
After the first such miss, Rooney swept home Antonio Valencia's cross before Juan Mata doubled the lead with a header which appeared to be offside.
Robin van Persie completed victory to keep United in third, eight points behind leaders Chelsea.
Liverpool's form has been dismal - three of their last four wins have been against bottom six clubs and they were knocked out of the Champions League midweek after a poor show against Basel. It's a far cry from a year ago when Liverpool were vying for top spot going into Christmas.
There have been sounds of discontent on Merseyside already and it will be interesting to see if the American owners will stand by their man.
It seems a risky supposition given that the Daily Star revealed on Friday Liverpool are taking a look at Diego Simeone as a possible replacement for Rodgers.
But owner John Henry has decided to be patient with Rodgers but only after he was allegedly talked out of wielding the axe by club legends.
Stan Collymore said: "I heard two weeks ago that John Henry has tried to sack Brendan Rodgers a couple of times already.
"He was persuaded not to by Ian Ayre (CEO) and a couple of club legends, who were sought out for their counsel.'
Sunday's result cannot have done the Rodgers cause any good.