Lambert converted from the spot in the 90th minute of the Barclays Premier League opening-day clash, three days on from heading the winner for England in their 3-2 triumph over Scotland with his first touch in international football.
At the age of 31, the former Bristol Rovers, Rochdale, Stockport and Macclesfield player is the man of the moment, and Pochettino could not be more delighted for him.
Regarding Lambert, scorer of 15 goals for Saints last term in what was his debut Premier League season, the Argentinian said: "He is having a dream week.
"Everything is going well for him - actually, everything is perfect for him.
"The greatest thing about this whole story about Rickie Lambert is that he deserves everything he is getting, all the success he is getting, because he is a brilliant person and a great footballer.
"We are very lucky to have him, not just as a player but also as a person, within our squad."
The penalty-taking prowess of Lambert is well known and asked if he had had any doubt over whether or not the Merseysider would score the spot-kick, Pochettino said: "If you review the images of when the penalty was taken, I was leaning against the wall, very calm and cool. I had no doubt."
As for the man himself, Lambert - whose third child was born on the same day he was called up for the Scotland game - said he was "overjoyed" with how the last few days had gone for him.
He told Southampton's official website after the Albion match: "It's a great way to finish the week.
"It'll be a week that I'll never forget in my life and to finish it off with a win for Saints is superb."
On his penalty, he added: "I tried to just whack it into that corner and thankfully, it's gone right into the corner because I think he (Baggies goalkeeper Ben Foster) has got fingertips to it as well.
"I knew I'd caught it sweet because it would have been a hell of a save, if he'd saved it."
Southampton deserved their three points having been the better side for much of the contest.
They had the ball in the net in the first half through Adam Lallana's close-range finish from a Lambert header, but it was ruled out for offside.
Referring to that incident, Pochettino said: "We actually scored a goal before (the penalty) that wasn't allowed to stand and that was a legitimate goal. We were unlucky with that decision by the referee.
"But I think in the overall reading of the game, we were superior to West Brom.
"You always have to believe you are going to win, through to the last whistle, and because of that I want to congratulate my team because even in the second half, when West Brom were pressing us quite hard, my team showed the resolve to keep it at 0-0 and we were able to get that winning goal in the end."
Pochettino said he had not seen replays of the incident that led to the penalty being awarded, where Youssouf Mulumbu was adjudged to have fouled Luke Shaw.
But West Brom boss Steve Clarke was sure what he made of it.
"I thought it was very, very soft, I have to say," Clarke said.
"I have seen it again a number of times on video.
"It was clever play from the Southampton player and to make that call at that stage of the game - I thought it was a very big decision (for referee Kevin Friend) to make."