Both finishes from Anthony Martial and Jesse Lingard were of the highest quality, but the build-ups were avoidable with the Blues carelessly losing possession for both.
The Toffees slipped to back-to-back defeats following a loss at Bournemouth two days ago, which ended a seven-game unbeaten run under Allardyce.
And the Everton boss felt his charges were the architects of their own downfall and admitted they cannot afford such errors given their lack of creativity.
Not enough goals in this Everton side, admits Allardyce
"Our attacking powers are limited, we know that," he admitted. "I knew that before I arrived here and that's why I've worked so hard on keeping clean sheets.
"That's going to be our main object to make sure we pick up more and more results and more and more wins, by not conceding a goal knowing that one goal can win us three points because our ratio of chances created is very limited.
"That's because we need to strengthen that front-line, we all know that. We're trying to do the hardest job of all by securing a player in that position. It wasn't achieved before the start of the season and it may not get achieved this January.
"We're trying but until that happens, we've got to maintain our defensive solidarity.
Allardyce was particularly frustrated by how his side failed to close down any of United's forwards in their lead up to their opening goal, which came from Wayne Rooney's misplaced pass on the edge of the visitors' box.
"I was very disappointed how we conceded the two goals today by really giving them the opportunity to score when we didn't need to," he continued.
"Both goals were down to our errors really rather than Manchester United's great play because when United pick the ball up just outside their half after we've given it away, they go from that end of the pitch to the edge of our 18-yard box with five and six of our players still behind the ball but not any of them six players deciding to go and challenge the ball or shut the opposition down.
"Ultimately in the end we end up with four of our players six yards inside their own box where they shouldn't be and by the time somebody decided to shut Martial down, he's put it in the back of the net. That was very avoidable in the first place, from basic defending.
"If I have to be super cynical, the player on the ball entering our half you would normally see a player foul him now and take the booking but we didn't even get near anybody."
Everton boss vows to deal with defensive issues
The 63-year-old hailed the combative display of midfielder James McCarthy from the bench and felt the Irishman provided the example his team-mates needed in the middle.
Yet he also acknowledged the need for his side to revisit their defensive principles on the training ground once they are the other side of a packed festive schedule.
He added: "We recovered from that with the substitutes. McCarthy, for me, was the shining light of the recovery because he actually went and tackled somebody and showed the rest of the team how we had to go about our job in shutting United down.
"We then worked our way back into the game and we looked a threat again, which we hadn't looked for that 20-minute, 25-minute period in the second half.
"The game is in the balance with 10 minutes to go, we have a throw in in our half and we throw it straight to United and United go and score. I don't know.
"Maybe it's fatigue, with all the games we've played, it's a lack of concentration. Certainly it will not go without me dealing with those sorts of situations when I get the opportunity.
"Maybe we've not been able to do it because we can't practice it and haven't been able to practice anything for the last 12 days with all the games.
"Maybe the lads have forgotten a little bit, so as soon as we can get Liverpool over with, we can get on the training ground and remind the lads how to go about their business and how to get better.
"Overall, I didn't think the performance was too bad apart from that 20 minutes or so in the second half. We've got to keep trying to improve and grow from here."