Despite the home side's best efforts, Villa held on for a point and could even have snatched all three at the death when Jussi Jaaskelainen saved excellently from Christian Benteke.
West Ham did have the best chances, however, with Kevin Nolan, Ravel Morrison and Joe Cole going close in a relatively even game.
Speaking to Sky Sports, Lambert was happy with a hard-earned away point, and indicated how far his side have progressed in just one season.
"We've always got goals in us. This is a big point for us, away from home," he said.
"Last year we might have lost this one, but the resilience and the attitude from the players is great.
"On chances created, I thought we had the better ones.
"It's a difficult place to come, but I thought we had the best clear-cut opportunities. Andy (Weimann) had a chance in the first-half and then Christian (Benteke) hit the bar and forced a fantastic save."
The game's controversial moment came in the first half as Andreas Weimann broke through the Hammers' backline, entering a foot-race with Morrison for the ball before falling to the ground, calling for a free-kick.
His cries went unheard, however, but Lambert's comments point towards what was a difficult decision to make for match referee Keith Hill.
"I just think that when the ball is on the safe side of you it's difficult for the opponent to win it fairly," he said.
"I thought there'd been a tug on it, but I've spoken to Keith and he calls it as he sees it."
Villa's two best opportunities to break the deadlock fell to striker Benteke, who first smashed the post with a header then nearly slotted past Jaaskelainen.
Fitness concerns still surround one of the Premier League's most-heralded marksman, having just returned from a hip injury, and Lambert was quick to lessen the expectations on the young forward's shoulders.
"He's not too far off. He's been out for a month, and it shows how much of a threat he is," he said.
"People get caught up in the moment. He's only 22 years of age, and he'll have his games where he doesn't score every time.
"The chances he creates and the way he plays, he's been incredible."
Austrian striker Weimann provided Lambert with a real injury scare in the second half after he came off with a hamstring strain, although the Villa boss had words of praise for his replacement, Libor Kozak.
"We need to see how he is, but I thought Kozak came on and did great."