Slack marking in the box allowed Cristiano Ronaldo the time and space to head The Red Devils in front before a Ruud van Nistelrooy brace sealed the win.
Harris stated the side had stuck well to player-manager Dennis Wise's game plan but were undone by giving United too big a sight of goal.
"We did what we could, had a game plan and stuck to it. It was disappointing to be 1-0 down at half time," explained Harris.
"We were punished for a mistake for the first goal and even though Ruud van Nistelrooy got two goals he had a quiet game. That tells you everything."
Millwall assistant boss Ray Wilkins was proud of his players, but slightly unhappy at the winning three-goal margin - as it flattered United.
"The players did themselves and the club proud. We are not too pleased with the scoreline," explained Wilkins.
"We worked extremely hard and are proud. United are a good side, they pass well.
"They were a little bit fortunate but the better side won. Our lads have learned a lot.
"It's been a great day. I'm sure the supporters have enjoyed it."
