Van Persie scored the only goal at Old Trafford with a first-half header against his old club, whose lead at the top of the Premier League has been trimmed to two points.
But the Dutchman was happy to spread the praise on an afternoon when defeat would have been a major setback for the hosts and manager David Moyes.
"It was a massive win," he told MUTV.
"I always like to think positively but if you think negatively for one second, we could have been 11 points behind, which would have been dramatic.
"It was a six-pointer. Now we are five and we are right in the mix.
"I don't think you can beat a team like that with just a couple of individuals playing well.
"You need to be strong as a team. That is what we did.
"Everyone closed everyone down.
"If you look at Patrice Evra, every header or 50-50 challenge, he went for it. The back four did it. Even the front players did it.
"That makes a difference."
Van Persie admitted it was not the cleanest goal of his career.
However, once he rose to meet Wayne Rooney's corner, all that mattered was the ball ended up in the visitors' goal.
"It was a good corner from Wazza," he said.
"I headed it but then it took a touch off my shoulder, then it went in.
"It didn't really matter whether it went in off my head or my shoulder. I am very happy with that head-shoulder goal."