The Scotland right-back has penned a deal until June 2027, with reports suggesting that the initial fee is in the region of £10 million ($18.7 million).
Patterson becomes the Toffees' second acquisition of the January transfer window after the arrival of Ukraine left-back Vitalii Mykolenko on New Year's Day.
Upon completion of the move on Wednesday (AEDT), the 20-year-old shared his delight at signing for the Premier League side.
"I'm over the moon to sign for Everton and looking forward to getting started," he told the Toffees' website.
"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 really wanted me here and that made my decision. When a club really wants you, it makes your mind up early on.
"Long-term, I want to nail down a starting position and win some trophies to make the fans proud."
Patterson also revealed that he has already spoken to club captain Seamus Coleman, who he will be competing with for a spot at right-back.
"I’ve had a good conversation with Seamus," he added. "I am really looking forward to learning from him and working together.
"That is why I am here: to learn and get better as a player and, hopefully, win things and do well for the club. If I am working alongside Seamus, that is definitely going to happen."
Patterson, who will wear the number three shirt, has made six starts and five substitute appearances for Rangers in all competitions this season, registering one assist.