After a hotly-contested and even opening period, the visitors began the second half on the front foot and opened the scoring through Anthony Martial's 20-yard effort.
The Blues rallied, buoyed by a sell-out Goodison Park crowd, following the setback but Jesse Lingard put the result beyond doubt with a strike from similar range to the opener with nine minutes remaining.
“We need to strengthen our frontline - I think we all know that," said Allardyce. "We're trying to do the hardest job of all by securing a player in that position. It wasn't achieved before the season and it may not be achieved this January, but we're trying.
“Until that happens, we have got to maintain our defensive solidity and I was very disappointed how we conceded the two goals by really giving them the opportunity to score when we didn't need to. Both goals were down to our errors, really, rather than Manchester United's great play.
“When they picked up the ball just outside their half after we've given it away, they went from one end of the pitch to the edge of our 18-yard box with five or six of our players behind the ball - but not any of those players decided to go and challenge the ball and shut the opposition down. Ultimately, in the end, we end up with five of our players six yards inside our own box where they shouldn't be and by the time someone 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."