Rooney's corner set up former Gunners striker Robin van Persie, whose first-half header proved to be the winner against an Arsenal side who suffered their third defeat from their last six matches.
Man of the match Rooney told Sky Sports 1: "We knew we'd have to win today at all costs. I think we defended really well and thankfully Robin got the three points and we got the three points, and I think it was a great effort from everyone."
Chelsea's home draw on Saturday was followed within 24 hours by shock defeats for Tottenham and Manchester City, meaning United have now closed to within five points of leaders Arsenal, who suffered their first league defeat since the opening day of the season.
Rooney added: "We knew we couldn't afford to lose today and we couldn't let Arsenal get 11 points ahead of us, and I think 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' Red Devils have won five of the last six games and remain unbeaten in their last nine matches in all competitions.
Rooney said: "It gives us a massive lift. 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, and we've got a big push then until the New Year and it's our aim to still be up there."
United boss Moyes admitted he was worried about captain Nemanja Vidic's head injury.
Just before half-time Phil Jones and Arsenal goalkeeper Wojciech Szczesny clashed heads, for which Szczesny seemed to come off the worst and Jones received a booking.
Serbia defender Vidic then suffered a head injury in a similar incident a few minutes later after he collided with his own goalkeeper David de Gea's knee, and remained on the ground for a number of minutes.
When he got up he was led down the tunnel holding his head and was replaced by Tom Cleverley at the start of the second half.
Moyes said: "He's gone to hospital. I've only just seen it myself.
"I didn't see how bad it was from the touchline, but now I've had the chance to look at it again it's a terrible sort of clash, it's an accident and hopefully he'll be okay.
"I'll find out more from the medics later on."
United recorded a goalless draw in their midweek Champions League clash at Real Socieded and Moyes was delighted to get back to winning ways.
He said: "This was a big result for us today. Arsenal have been in terrific form this season and we had to play well ourselves today to get the result.
"I thought we started the game well and we were very good in the opening 20 to 30 minutes of the game. And I thought, in the end, our whole determination of effort, commitment and good play helped us in the second half when we were under more pressure."
Moyes believes the Premier League title will be a close call this season, and added: "I don't think anyone's going to run away with it.
"I think there are a lots of ups and downs, it's a close-run league this year.
"I think there will be a few shocks and we've think we've had one or two at the early part of the season, but I'm pleased we've been able to show a little bit more consistency and a bit more momentum, and let's hope we continue that."