The Dutchman scored his 12th goal of the season just 33 seconds into United's encounter with West Ham at Old Trafford.
It proved enough to give Sir Alex Ferguson's side a nervy win and consolidate their status as league leaders, with the Manchester derby against City looming next month.
"Manchester United have lots of players who will have good spells and can come up trumps when they need them to," said the West Ham manager.
"Robin van Persie is a quality player and we have largely kept him quiet.
"But he will score 20 goals for Manchester United, maybe 30. In the end that is normally the difference."
Allardyce could take plenty of positives out of his first visit to Old Trafford as a visiting manager since he presided over a 7-1 defeat for Blackburn two years ago.
With a side lacking any orthodox wingers, Van Persie checked inside from the left flank, then made his way to the edge of the area as Michael Carrick exchanged passes with Anderson.
Van Persie swivelled past Winston Reid as he collected Carrick's forward pass, then profited from James Collins' desperate attempt to block, which only sent the ball looping over Jussi Jaaskelainen and into the net.
Sloppy defending from Manchester United almost allowed West Ham back into the game, however, with Andy Carroll and Carlton Cole both coming close.
"It is a big problem to concede after 30 seconds," said Allardyce. "But our response was very good.
"Whether Manchester United played to their potential or not doesn't matter does it? They still won. They have a nasty habit of doing that."
Sir Alex Ferguson was not too unhappy with the performance, even though, adopting his diamond midfield, United failed to penetrate their opponents and Wayne Rooney was a particular disappointment.
"We got a great start," said Ferguson.
"You hope to go on from a start like that and some of our football was very good but we never quite finished them off.
"We have played with the diamond quite a few times this season and done very well so I don't think that is an excuse.
"But we have got the three points and are still ahead in the league.
"With Chelsea drawing it draws us away from them a little bit further.
"But, of course, City are on our tails so it is going to be a long season."
The main plus point from the evening was the performance of Brazilian fullback Rafael, who is having an excellent season and was making his 19th appearance.
"He was excellent tonight," said Ferguson.
"He came here as a young kid and was in the first team at 17 or 18.
"Of course he made a few mistakes because he is such an enthusiastic boy. But he is maturing and learning with that."
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