Koeman, who counts Ajax and Barcelona among his clubs as a player, brings his fellow former Dutch international and elder brother Erwin as his assistant at Southampton.
"I am thrilled to be the manager of Southampton Football Club," Koeman said in a statement. "From my very first meeting with (director) Les Reed, I could see clearly that this was the right club for me.
"We share the same philosophy of football, and the same belief in how to develop the club. The opportunity to continue the development of this famous football club was simply too good to turn down. I cannot wait to get started."
The Saints, who only five years ago were in the third flight of English football, had been seeking a new manager since Pochettino left for Spurs' greater spending power, and many commentators suspected the Argentine's departure would arrest the club's ascent.
BIG AMBITIONS
The club's ownership maintained silence as supporters railed against the manager's departure and a growing number of big clubs circled above Southampton's World Cup players, including Rickie Lambert, Adam Lallana and Luke Shaw for England, plus Croatia's Dejan Lovren and French midfielder Morgan Schneiderlin.
Lambert signed for Liverpool just before the World Cup and question marks remain about the others.
Koeman's appointment should settle the mood at St Mary's, however, and reassure players that the club, which finished eighth in the Premier League last season, have big ambitions.
His appointment was welcomed by former Southampton striker and club talisman Matt Le Tissier, who said on his Twitter feed: "Seems like an excellent (appointment)... big name to attract players."
Reed said the club had identified Koeman as their priority as soon as it became clear Pochettino wanted to leave.
"We have been following the development of Ronald's coaching career for some time and, through a more formal process this month, immediately identified him as our number-one choice.
"From the moment we met with Ronald, it was clear he had the vision and ambition to take our club forward, that he shared our football philosophy and could see the opportunity to move our club to the next level.
GLITTERING DUTCH TEAM
"The club, the players and the fans will all be excited to have a man of Ronald’s experience at the very highest level joining our club."
Koeman, who played in the glittering Dutch team who won the 1988 European Championships alongside Marco van Basten, Ruud Gullit and Frank Rijkaard, spent three successful seasons at Feyenoord who were the league's leading scorers last term.
He has also managed Ajax, Benfica, PSV Eindhoven, Valencia Vitesse Arnhem and AZ Alkmaar.
Koeman won the European Cup with PSV as a player in 1988 before joining compatriot Johan Cruyff's Barcelona 'dream team' that won four successive Spanish league titles and the 1992 European Cup.
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