Weimann held his head in his hands after shooting wide from six yards out with the goal at his mercy after Hammers keeper Jussi Jaaskelainen had spilled Christian Benteke's low shot.
It looked like proving a costly error in Villa's fight for Premier League survival until second half goals from Benteke and Charles N'Zogbia secured a 2-1 win.
But Lambert's faith in Weimann, Villa's second top scorer with nine goals, is unwavering after his side climbed out of the bottom three for the third time this season.
Lambert said: "I thought we should have had a goal in the first minute when Andi missed a chance and everyone was a bit hurt by that.
"It probably took Andi by surprise himself that he's missed it. He is one of the most natural finishers at the football club
"Heads could have gone down after the early miss but Andi Weimann has been brilliant for us this season.
"He has pulled us out of a few holes a few times and he will come back stronger than ever from that.
"He's only a young lad, people forget that. It's important he bounces back from it.
"One thing Andi doesn't lack is confidence, that's for sure. His effort and enthusiasm will stand him in good stead throughout his career."
Lambert knows what might have been had his side been more clinical in closing out games but insists the Villa players are in a positive frame of mind for the final three months of the season.
He said: "Our mindset is strong as we approach the crucial final third of the season.
"We've grown together as a team and from day one, we've maintained a really upbeat attitude.
"We know what we have to do. We have 12 games left and every one is a cup final.
"I am glad they saw it through against West Ham because they've taken a lot of knocks and deserve the credit coming their way.
"We had a two goal lead against Manchester United, West Brom and Everton.
"If we'd got two more wins, we'd be sitting in mid-table and that's the reality of it.
"But what we've started to do now is pull a few teams above us back into the mix.
"We have been playing well for the last couple of weeks, especially the second half against Newcastle, the first half against West Brom and at Everton."
Skipper and central defender Ron Vlaar said: "We are in a tough position and we have to work really hard to get our points.
"Last week at Everton, we didn't play that bad but we didn't get the result we wanted so the win over West Ham will bring a lot of confidence.
"It was very hard to keep in front but we did very well. We were a little bit lucky as well but we deserved some luck.
"Towards the end it was all long balls and battles, but the only thing that mattered was that we got three points and you could feel the relief, with the fans as well."
Premier League To Stay Up 12/13 To Stay Up: Aston Villa 1/100
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