The Scotland international has signed a four-year contract with the Saints and will link up with his new teammates later this week.
"Southampton seems a really good fit for me. Watching from afar, I really enjoy the attractive football that they play and I'm very happy to be here," Armstrong told Southampton's official website.
"Every season, I put demands on myself to create and score goals. I hope going forwards from here that it will be the same this term at St Mary's.
"I'm looking forward to meeting everyone and I'm very excited to get started. It's a four-year contract; there will be a lot to learn and I want to hit the ground running."
Southampton did not disclose how much they paid for Armstrong, but previous reports suggest that the fee could rise to £8m.