Former Arsenal player Robin van Persie scored the winning goal in their 1-0 triumph at Old Trafford to close the gap on the Premier League leaders to five points.
Chelsea, Tottenham and Manchester City all dropped points over the weekend to make United's three-point haul even more important and man-of-the-match Rooney was pleased with a job well done.
"We knew we'd have to win today at all costs," he told Sky Sports.
"We defended really well and thankfully we got the three points, it was a great effort from everyone.
"We knew we couldn't afford to lose today and we couldn't let Arsenal get 11 points ahead of us.
"With the other results today and yesterday, we knew a victory would put us right back in there and we're in a good position.
"We worked on free-kicks and corners as Arsenal have a lot of smaller players in the team and that could give us the opportunity to get a goal, and thankfully it's paid off."
David Moyes's side have now won five of the last six matches and remain unbeaten in their last nine matches in all competitions.
"It gives us a massive lift," added Rooney. "It's a great weekend for us going into the international games and we can all go off and play for our country and then come back.
"We've got a big push then until the New Year and it's our aim to still be up there."