The England international has been the subject of long-term interest from the south coast outfit, who are thought to have snapped up the shot-stopper for a fee of around €12.5 million.
Speaking after his side's 1-0 friendly defeat to Bayer Leverkusen on Saturday, Saints boss Koeman confirmed that the move for the former Newcastle United man was imminent, and has now shared his delight at signing the 26-year-old.
"Fraser is a very talented player and we are delighted to be able to bring him to Southampton," Koeman told the club's official website. "This is another important part of our rebuilding process ahead of the start of the season.
"Fraser is the perfect fit to help complement the players we already have, and he will add a lot of quality to the squad.
"At 26, he will enjoy the best years of his career at Southampton."
Forster has made almost 200 appearances for Celtic in four years, earning a first England call-up in October 2012 before making his international debut against Chile in November last year.
He could be joined at Southampton by Leeds United right-back Sam Byram, with Koeman confirming interest in the defender - though an initial bid has yet to be accepted.
"We are after that, we lost [Callum] Chambers to Arsenal, it was a player we had on the list and it's true that we had a bid for him, yes," confirmed the Dutch boss. "What I heard was that Leeds didn't accept it."