The 27-year-old Iceland midfielder has put pen to paper on a five-year contract and becomes the Toffees’ eighth summer arrival.
“Hopefully I will continue to create goals and score goals for the team and contribute to the kind of success we’re seeking at the club,” Sigurdsson said.
“This is an ambitious club and it is clear we are moving in a really good direction.”
Sigurdsson’s proposed move has dragged on for much of the summer, with Swansea initially valuing their star man at £50m (€54.7m).
Finally securing him now, at a slightly lower price, is set to take Everton’s spending since the end of last season to over £140m (€153.3m).
Koeman, who has already this summer brought in the likes of Jordan Pickford, Michael Keane, Davy Klaassen, Sandro Ramirez, and Wayne Rooney, said of Sigurdsson: “We worked a long time to get this deal in and he was one of the key players to bring in.
“We knew we would lose (Romelu) Lukaku for this season, with that we lost 25 goals, and we need players who will bring in productivity to the team.
“He is that type of player, with a lot of productivity.
“He knows the Premier League and he had really good seasons with Swansea.
In my opinion he is, in his position, one of the best in the Premier League.”
Sigurdsson will not feature in tonight’s Europa League qualifier first leg against Croatia’s Hajduk Split (8.05pm, Goodison Park), but Koeman has indicated he could make his Everton debut in Monday’s league clash with Manchester City at the Etihad Stadium.
Koeman has said Everton are waiting to learn the severity of an apparent hamstring problem suffered by Ross Barkley.
Reports have suggested Barkley, whose future with the club remains unclear, is facing six weeks out due to a hamstring tear.
Asked about that, Koeman said: “He stopped training, I think it was last Monday. It looks like a hamstring injury but we need to do more tests today and tomorrow to know exactly what the injury is. We need to wait.”
The Dutchman added that there was “no update” in terms of the 23-year-old’s “situation”, and when asked if he was in his plans for the coming season, he said: “I will wait until the end of August.”
Barkley, who is entering the last year of his contract, appears destined to leave the club having turned down the offer of a new deal.
He has been linked with a move to Tottenham, but Koeman said last week there had been no offers for the England international.
Koeman has also signalled he would still like to strengthen in the current transfer window beyond the capture of Sigurdsson.