The 32-year-old Denmark international was left out of Arsenal's squad for Wednesday's friendly against Real Betis as the move was finalised, having made just 20 appearances in all competitions last season, only seven of them in the Premier League, despite helping the club to a first title in 22 years.
He arrives at Goodison Park having spent six seasons at Brentford, where he made 196 appearances, 122 of them in the Premier League, scoring 13 goals and registering 18 assists after captaining the club through one of the most successful periods in its history; he had joined Arsenal for an initial £10m last summer before the move to Everton was struck for roughly £7m.
Speaking exclusively to evertontv, Norgaard said he was ready for a change after limited game time in north London.
"It feels amazing. I'm so, so happy and pleased to have joined Everton," he said. "There were many factors to it. I felt I needed a change and to play a bigger football role.
"When I spoke to David Moyes and others at the Club involved in the transfer, I heard about the whole project and I was very intrigued."
He also spoke of his admiration for Everton's atmosphere, both at Goodison Park and the club's new stadium.
"I just remember the ground was shaking – literally shaking. Then we came to the new stadium last year and I was so surprised because you often see clubs lose that intimacy and atmosphere. But I was so overwhelmed by how good the atmosphere was as well. So I can see that the soul from Goodison has been brought to the new stadium."
Norgaard also revealed he had sought the view of Arsenal manager and former Everton midfielder Mikel Arteta before making the move.
"He had only positive things to say about Everton and also David Moyes," Norgaard said. "He played for him for a long time and the whole experience he had as an Everton player seemed to me as if he had a fantastic time here and really also excelled as a player under David Moyes."
Norgaard, who is from Copenhagen, rose through Lyngby's youth ranks before a stint in Germany with Hamburger SV, then returned home to captain Brøndby to a string of domestic honours; a season in Italy with Fiorentina followed before he made the move to English football with Brentford. He becomes Everton's fourth signing of the summer, following Hayden Hackney, Tyrique George and Merlin Röhl through the door, as David Moyes looks to fill the void left by Idrissa Gueye's departure.
Discussing what he can offer the squad, Norgaard was confident his leadership would translate seamlessly.
"I think leadership is something I have to bring and it's something I will bring without really thinking about it," he said. "I think the Club is moving in the right direction and last year was proof of that. I thought there were so many good players in the team and Everton was so difficult to play against. There is a lot of optimism around the place. I can't wait to join in on that."