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.
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 when 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."