Wanyama, who revealed he had been won over by the south-coast club's dogged determination to land him, had already said his farewells to his Celtic team-mates on their current German tour before jetting off to seal the bumper deal.
"I am happy to be part of Southampton's squad," said the powerful midfielder. "It means a lot because it has been my dream to play in the Premier League and now I am here.
"Finally it has happened and we've come to the end of negotiations, so I am delighted. There were many clubs that inquired about my services over the past season, but it's Southampton who kept on trying to acquire my services, and that trust is very important.
"Southampton is a good club and it has good players, so I just want to train hard and work hard to be a part of the starting XI. We'll see where that will take me. I'm very grateful to be here.
"I've seen that the stadium is great and the dressing room is looking good, and I know some of the players. I hope they will help me and I think I will get along with the others, so it will be good."