The Iceland midfielder has agreed a five-year deal to join for an undisclosed club-record fee.
“He was a great player, one of the top players of his time,” Sigurdsson told evertonfc.com. “Now he is a really good manager and his vision for Everton is clear."
Koeman added: “In my opinion in his position he is one of the best in the Premier League."
Sigurdsson, who has also played for Tottenham Hotspur, has made 182 career PL appearances. He has recorded 42 goals and 33 assists.
The 27-year-old could make his Toffees debut at Manchester City on Monday 21 August.