The Saints completed the signing of the Scotland international from Celtic earlier this week, and the 26-year-old has penned a four-year contract at St Mary's.
Rodgers has said that Armstrong's 'dream was to play in the Premier League', and the Hoops manager is pleased that the midfielder has moved to England despite admitting that losing him is a blow to the Scottish champions.
"I am really happy for him. It was always his dream to play in the English Premier League," Rodgers told reporters.
"He has gone to a fantastic club that is very, very well run and very well organised. They have got a great coaching environment down there with Mark Hughes, Eddie [Niedzwiecki] and Mark [Bowen] and all the staff that they have.
"They have got a wonderful training facility. The ground is a very good ground so I think it is a great fit for him and I think he will do very, very well.
"Of course you will be sad to lose good players but from the club's perspective they have done good business and Stuart achieves his dream so I am really happy for him."
Armstrong could make his Southampton debut in their first pre-season friendly against Schalke 04 on July 5.