The hosts dominated the Premier League contest for long spells at the Emirates Stadium before falling behind to a Kieran Gibbs own goal in the 56th minute.
Rooney streamed forward to seal the points and Angel di Maria should have added a third on the counter-attack before Olivier Giroud netted midway through nine minutes of injury time as Arsenal battled in vain to stop United claiming a first away victory of the season.
"It's a very important victory," Rooney told Sky Sports. "We weathered the storm at times, especially first half.
"Our goalkeeper [David de Gea] was fantastic today and we hit them with two good goals.
"It was a different game. They had possession and we hit them on the break - that was our gameplan."
United's lengthy injury list led manager Louis van Gaal to select a back three featuring rookie centre-backs Tyler Blackett and Paddy McNair, and Rooney felt the youngsters performed well in a determined team display.
"We've put five at the back and a bit of experience in myself, Michael [Carrick] and [Marouane] Fellaini in the midfield to try and protect them," he added.
"I felt we did that well at times and we knew we had legs in terms of Robin [van Persie], Angel and myself joining in on the break. We saw that to good effect in the second half.
"I think it's a massive result for us. Two very young players, Paddy McNair and Tyler Blackett, in such a big game - they've coped well.
"A clean sheet - I'm sure the defence and the goalkeeper will be delighted with that.
"This will give us a lot of confidence, a lot of belief and it's come at the right time for us."
Despite a stuttering start to the Van Gaal era, United have now broken into the Premier League's top four as they bid to reclaim the coveted UEFA Champions League spot they let slip during a shambolic campaign last time out.
"We'll take each game as it comes," Rooney added. "It's been an indifferent start to the season, but we hope now that we can go on a run of games and enjoy the position we're in after Christmas.
"We want to try and build a bit of momentum going into the new year so today was a massive result in terms of starting that."
