Sigurdsson, 27, has signed a five-year deal with Ronald Koeman's men and is likely to be unveiled during his new side's Europa League play-off round clash against Hajduk Split at Goodison Park on Thursday (17 August).
The confirmation of the deal brings to an end over a month of posturing and infuriating negotiations. Swansea had initially placed a £50m price tag on the talismanic playmaker during the embryonic stages of the summer transfer window, but Everton were reluctant to go so high for the former Tottenham attacker, who dragged Paul Clement's side away from the relegation mire last summer.
Clement said there was a "difference of opinion" over the fee last weekend but a deal was finally agreed on Tuesday, with Everton paying a set amount of £40m with £5m potentially to be worth in add-ons.
Koeman has admired Sigurdsson for a long time and made him his top target this summer, and is delighted to have finally captured the former Reading academy product: "I'm really happy because the Board and especially our Chairman, Bill Kenwright, did everything to get him. It took some time but finally we have signed the player," Koeman told evertontv.
"In my opinion, in his position he is one of the best in the Premier League. We have the player here and that's really good for the team."
Everton supporters may have to wait to see Sigurdsson make his debut for the club. The talented forward played just 45 minutes of football during pre-season and was omitted from Swansea's pre-season tour of America due to the speculation that surrounded him.
Sigurdsson may play some part against Manchester City on Monday, but for now is just delighted to have finally joined Everton: "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 told evertontv.
"This is an ambitious club and it is clear we are moving in a really good direction. The most important thing is that this team keeps winning games and climbing the table - and does that throughout the season.
"For myself, I would like to score as many goals as possible and create as many goals for my teammates as I can. These are my targets. But it is always the same thing: as long as the team is winning I will be more than happy."
Sigurdsson is Everton twelfth signing of the summer, but Koeman is keen to bring a couple more players before the end of the transfer window. The former Southampton boss is looking for a centre-forward and left-sided central defender, while a winger may sneak onto the Dutchman's agenda.