The Iceland international has signed a five-year deal with the Toffees, keeping him at Goodison Park until June 2022.
"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," he told Everton TV.
"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 considered one of the most creative players in the Premier League, having been directly involved in 21 top-flight goals for the Swans last term.