The England international pounced just before half-time before three own goals and a strike by Dimitar Berbatov took United to 56 points.
Liverpool overcame the dismissal of Sotiros Kyrgiakos to beat Everton 1-0 in a fierce Merseyside derby at Anfield to move into the top four for the first time since October.
Dirk Kuyt headed the winner 10 minutes after halftime.
Tottenham Hotspur slipped to fifth in the table after a goalless draw at home to resilient Aston Villa.
Spurs created most of the chances but could not find a way to break down one of their rivals for a Champions League qualification place.
Tottenham have 43 points from 25 games, leaving them a point behind Liverpool and six adrift of Arsenal.
Villa stay seventh with 41 points from 24 games but are now level with Manchester City, who suffered a surprise 2-1 defeat at Hull City.
OWN GOALS
Portsmouth weathered the inevitable early onslaught from champions United at Old Trafford but came unstuck after 40 minutes when the irrepressible Rooney netted his 21st league goal of the season.
The south-coast side conceded another just before halftime when Anthony Vanden Borre turned a Nani cross between his own posts.
Carrick struck a deflected shot past England goalkeeper David James, Bulgarian Berbatov atoned for a first-half miss with an excellent long-range strike and Marc Wilson helped Patrice Evra's pass into his own net to complete the rout.
"It took us a while to get the first goal but we had a lot of pressure, a lot of possession and we were patient enough," United mananger Alex Ferguson told MUTV.
"I think that's important, especially when teams come and sit there, like they did today. We were fortunate with the second one, but that was important. It opened the game for us and in the second half we had a good peformance."
Big-spending Manchester City slipped up in their pursuit of fourth place when they lost at lowly Hull, leaving them three points behind Liverpool albeit with two games in hand.
Visiting manager Roberto Mancini was left rueing his side's failure to match Hull's determination and said they must change their mental approach on the road to secure a top four finish.
"Hull were more aggressive but if you understand why you lost sometimes it's better. We must change our mentality and play away games like we play at home," the Italian told Sky Sports.
Burnley battled to a 2-1 win over West Ham United to take them out of the relegation zone, Stoke City beat Blackburn Rovers 3-0, Bolton Wanderers and Fulham played out a 0-0 draw and struggling Sunderland and Wigan Athletic drew 1-1.
