The Hammers fell two goals behind with just 22 minutes on the clock at Old Trafford, but despite Diafra Sakho pulling one back for his side before the interval, the visitors proved wasteful in front of goal to see a point pass them by.
"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," Goal.com quotes Allardyce as saying.
"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."
West Ham, who are currently 11th in the Premier League table, return to action next weekend when they take on Queens Park Rangers.