The Dutchman scored his 12th goal of the term just 33 seconds into Manchester United's match with the Hammers at Old Trafford last night.
It proved enough to give the Red Devils a nervy win, which allowed them to consolidate their status as league leaders, with the Manchester derby with City looming at the Etihad Stadium on December 9.
And with the championship already looking like a two-horse race as Chelsea crumble, Allardyce thinks Van Persie could be a key factor.
"Manchester United have lots of players who will have good spells and can come up trumps when they need them to," he said.
"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 boss since he presided over a 7-1 defeat for Blackburn Rovers two years ago.
When Van Persie struck so early, the Hammers might have feared something similar.
In the end, though, they were the ones who finished stronger, only for Carlton Cole to waste their best chance when it was saved by Anders Lindegaard.
"It is a big problem to concede after 30 seconds," said Allardyce. "But our response was very good.
"We didn't create much but we went looking for the equaliser and the best chance fell to Carlton Cole and he didn't quite catch it.
"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."
Meanwhile, Manchester City midfielder Gareth Barry was pleased to grind out a 2-0 win at Wigan to stay within a point of United.
After a poor first half, the Premier League champions took control with Mario Balotelli scoring his first league goal of the season and James Milner chiming in with a screamer.
The goals came after some City fans had jeered boss Roberto Mancini's decision to swap Sergio Aguero for Aleksandar Kolarov.
"We found it quite tough going in the first half, we had some stray passes but we ground it out in the second half and it's a great three points," admitted the England midfielder.
"Credit to Wigan, we had to change things to get past them.
"When the game is going our way the manager is the best tactician there is, that's the world we live in."