Pochettino was then named manager of the month for October on Friday, before leading his side to a stylish 4-1 victory against Hull on Saturday afternoon.
Morgan Schneiderlin, Lambert, Lallana and Steven Davis netted in a performance that will have left watching England manager Roy Hodgson delighted.
"I am very happy," Pochettino said. "I want to congratulate all of my players. They put in an amazing performance - all of them.
"They had a great performance and I am very pleased with all of them.
"I hope all the players that got called up to their national sides get the chance to play. It would be great for them.
"I think they all have amazing potential, they just all need to keep on developing their football, they all need to keep progressing as football players and playing for their national sides will give them that confidence and chance to perform at a higher quality.
"We just need to keep on for Southampton working with the same humility and the same work ethic."
Asked if this had been one of the best weeks of his career despite Wednesday's Capital One Cup exit to Sunderland, Pochettino said: "I would say it was one of the best, but unfortunately last night I wasn't able to sleep at all because I had a cold.
"Professionally it has been one of the best weeks I have had but on a personal note I'm not doing so well."
The win against Hull was Saints' fourth league win in succession, making this their best ever start to a top-flight season, trumping the 1983-84 campaign when they finished runners-up.
Lallana has been key to that success and produced another man-of-the-match display against the Tigers, as well as an exquisite solo goal.
Picking up the ball midway through the opponent's half, the Saints captain drifted to the left, rode the challenge of Curtis Davies and ghosted past Ahmed Elmohamady, before slotting home right-footed past Steve Harper from an acute angle.
"I am not surprised at all," Pochettino said. "Ever since we arrived in January I saw how well he trains every day, how hard he works every day.
"Little by little he is showing the amazing football he has inside. He has a great talent and is an amazing player."
Lallana was one of six English players in the Saints starting line-up - a home-grown focus praised by Hodgson when he announced the senior squad on Thursday.
Asked if the England boss had thanked him, Pochettino said: "I have not got a chance to see him yet.
"There was some traffic before when we were going to see each other so it didn't happen in the end.
"He knows of my admiration for him and I am the one who admires him for his work."
Pochettino is the one with a growing list of admirers, though, and Hull boss Steve Bruce admits Southampton are a growing force in the Barclays Premier League.
"Sometimes you have to hold your hands up and say we were well beaten," said the City manager, who saw Yannick Sagbo net in a much improved second-half display.
"At least in the second half we showed some character and restored some pride.
"Southampton are the best side in the division outside the usual suspects challenging for the top four by a country distance. They have very good players.
"The better team on the day by a million miles won the game.
"Maybe Southampton are not a small club anymore and maybe they do have enough to compete at the top end of the table."
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- Adam Lallana
- Rickie Lambert
- Roy Hodgson