Rangers have received a club-record fee for the 20-year-old Scotland international, which the PA news agency understands is £11million plus add-ons.
He becomes a second signing of the transfer window for the Toffees, who are currently 15th in the Premier League, with Ukraine left-back Vitaliy Mykolenko having already arrived from Dynamo Kiev for £17m.
Rangers academy product Patterson made his senior debut for the Glasgow outfit in January 2020 and has played 25 times for them in total.
Patterson, holder of six international caps, told evertontv: “I’m over the moon to sign for Everton and looking forward to getting started.
“It was an easy choice to sign. This is a massive club with huge history and the fanbase is tremendous.
“Obviously we aren’t in the place where we want to be right now but we can definitely get there with the talent in the changing room and staff here, and I have full belief we can do that.
“It is really exciting to come to a club that has huge talent in the team. I am going to work my socks off and hopefully I can do well with them in training and in games.
“It was also great for me knowing the manager (Rafael Benitez) really wanted me here and that made my decision.
It was an easy choice to sign. This is a massive club with huge history and the fanbase is tremendous.
“When a club really wants you, it makes your mind up early on. I wanted to come here from the very start. I’m delighted to be here and really looking forward to playing in front of the fans at Goodison.
“Long-term, I want to nail down a starting position and win some trophies to make the fans proud.”
Patterson said Evertonians could expect “athleticism, desire and strong challenges, along with good forward play” from him.
He also said he had spoken to the club’s skipper and fellow right-back Seamus Coleman, adding: “I am really looking forward to learning from him and working together.”
Patterson helped Rangers win the Scottish Premiership last season, and his appearances for them have included nine in the Europa League.
Speaking about the deal in quotes on Rangers’ official website, the club’s sporting director Ross Wilson said: “I have been in discussions with Everton chairman Bill Kenwright for a number of weeks. It has been a pleasure to deal with Bill – a man who represents his club with genuine class.
“It is true that there were previous discussions last summer but we were always clear that in our model players will only leave at the right time and for the right number.
“We have now established a significant club record and there will be even further financial incentives for Rangers as Nathan’s Everton career evolves.
“We were very clear in these discussions that Nathan is a player and person who we hold in the highest regard and the deal had to represent a club record for Rangers.
“Our academy staff can be proud of the role they’ve played in Nathan’s development. We are proud of Nathan, he leaves as a champion and I will follow every step of his career and journey and he will always be connected to Rangers. I am sure we will see him back supporting the lads at Ibrox soon.”