The visitors fell behind in the fifth minute on Saturday when Wayne Rooney netted his 176th Premier League goal to become the third-highest scorer in the division's history.
Midway through the second half, West Ham conceded again as Alex Song was robbed of possession, allowing Falcao to set up a goal for Robin van Persie.
West Ham fought back and halved their deficit through Diafra Sakho, but wastefulness in front of goal meant they were unable to claim an equaliser despite the 59th-minute dismissal of Rooney.
"If we hadn't been so stupid in the first place by gifting them two goals I wouldn't be sat here being as frustrated as I am," said Allardyce at his post-match media conference.
"Our sloppy start was just something I didn't expect.
"Manchester United have got gifted players and if you make silly mistakes like we did they will punish you, and they punished us twice, especially the second one.
"We came back from that and scored from a corner and had several chances to score more goals but I'm afraid our quality of finishing was not good enough on the chances we created.
"[At] both ends we've had enough chances to score more goals but we fluffed them, we blew it."
Kevin Nolan looked to have levelled the score with two minutes remaining, only for his effort to ruled out for offside, much to the frustration of Allardyce.
"Somebody's suggesting his [Nolan's] head was offside," added the West Ham boss. "If he [the assistant referee] can see that his head is offside then he is a super human being.
"I thought it was onside, there's no doubt about that in my mind, that would have got us a 2-2 [draw] but really it's our own fault today.
"Referees make mistakes; we have to accept that."
