Robin van Persie struck a first-half hat-trick as United clinched the Premier League title for 2012/13 but Lambert was happy his players could limit the damage on a tough night at Old Trafford.
"I knew it was going to be really, really tough," Lambert told Sky Sports. "But we kept on going and it was important we did not capitulate like we did at Stamford Bridge earlier this season, but congratulations to Manchester United.
"I am proud that my lads kept on going, credit to them, especially against the best team in the country.
"You want to give a good account of yourselves at Old Trafford but after the start we had you are not wanting to concede more goals. The lads kept on going and, for us, next Monday [against Sunderland] is the one we have to win.
"Manchester United have worked really hard for each other, they have a winning mentality and the best team always wins the league."
The loss kept Villa in 17th spot on the table with four matches remaining but they sit three points and goal difference ahead of 18th-placed Wigan Athletic.