Villa led 2-0 courtesy of two goals from Andreas Weimann before a typical United comeback enabled them to triumph 3-2 at Villa Park on Saturday.
But Lambert had already seen signs in recent weeks of a squad unproven at top-flight level starting to make their mark and feels that was confirmed with their showing against the Red Devils.
He said: "I'm proud of them, what they've done. It's a young team growing up and they'll take a bit of belief from that (against United).
"The experience will make them better and Manchester United have come back like that against many an established and experienced team.
"People thought United would come here and roll over us pretty easily but we gave it a right good go and I've told them 'You've got nothing to be down about.'
"I think we have been playing well for the last few weeks and we can take a lot of belief from that performance.
"The young lads will make mistakes like everybody else but if they keep showing that enthusiasm and that hunger for the game, then they and the club will be fine."
Lambert, who also put the accent on youth at his previous club Norwich, added: "The players can take great confidence from leading against Manchester United.
"It will make them grow a bit. You can see they are a young side but you wouldn't have thought that the way they played.
"They played with a lot of maturity, a lot of discipline in their game and I can't fault them for the way they went about it."
Striker Christian Benteke, 21, is already creating a favourable impression and caused constant problems for the United defence.
The Belgian international made a £7million move to Villa from Genk during the summer.
Lambert said: "I think he's not just a strong player, his technique is terrific and for someone who is just 21 he can be anything he wants.
"Christian keeps going. He's a brilliant guy as well. His character is terrific but my God, he's a proper number nine. He did give the United centre-halves a run for their money."