Robin van Persie struck a hat-trick inside the first 35 minutes at Old Trafford to effectively put the match beyond Villa, as the Red Devils went on to secure their 20th Premier League title.
Lambert's men suffered an 8-0 humiliation against Chelsea earlier this season, and the Scot has applauded his side for showing resilience and avoiding a repeat result.
"I am proud that my lads kept on going, credit to them, especially against the best team in the country," Lambert told reporters.
"I knew it was going to be really, really tough 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.
"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."
The end-of-match statistics showed that Lambert's men enjoyed 48% of possession and had the same number of shots on target as the hosts.