The Hammers were undone by goals from Danny Welbeck, Adnan Januzaj and Ashley Young at Old Trafford and failed to capitalise on their midweek win at Tottenham
Allardyce's men failed to repeat their mid-week heroics – which saw them score twice in the last 10 minutes to win their Capital One Cup quarter-final at Tottenham 2-1 – and were second best throughout at Old Trafford.
Danny Welbeck and Adnan Januzaj took advantage of poor defending to score for United in the first half before substitute Ashley Young added a third with 18 minutes remaining.
Carlton Cole pulled back a late consolation, but Allardyce had no complaints with the result as his side remain 17th in the Premier League table, just one point above the relegation zone.
"It is the right result, wasn't it? I can't complain about the result," Allardyce said.
"I can't complain about the fact that we were 2-0 down at half-time because of our deficiencies in our performance.
"Manchester United's players will take full advantage of basic mistakes in your own half, which they did on the two goals they scored before half-time.
"It's always a tough test from there on when you're here. My disappointment was that the first half was not good enough from our point of view."
Allardyce kept faith with Spaniard Adrian between the posts after he impressed at Tottenham in place of veteran Jussi Jaaskelainen and defended his team selection despite the goalkeeper's poor positioning for Welbeck's opener.
"At the end of the day, we have to find a replacement for Jussi," he said.
"We didn't need to take Adrian from Spain and throw him straight in [after signing him], which we thought would be the wrong thing to do based on Jussi's performances last season and certainly this season.
"I could argue he was a little unfortunate to be left out, Jussi, after what he has done recently, but [Adrian put in] such a good performance on Wednesday night, and I think he made some good and vital saves in the early part of the game."
The West Ham boss is looking forward to an improvement in his side's injury problems, with the likes of Andy Carroll, Stewart Downing and Winston Reid still sidelined.
"I've got to go all out to get some of the players who are injured – Kevin [Nolan is] back at least from suspension now – and get them fit and back in the team," he added.
"Because they will make a difference to us. We have too few players who are fit at the moment for me in this period of time."