The midfielder joined the Toffees in a £20 million deal on Thursday.
Schneiderlin moved to Goodison Park in a deal worth an initial £20 million on Thursday to become the club's second signing of the January transfer window after Ademola Lookman.
The 27-year-old France international had fallen heavily out of favour at Old Trafford under Jose Mourinho, and the most recent of his three Premier League appearances this season came back in mid-November.
He has now reunited with his former Southampton manager Koeman, under whom he played for one season for the Saints before his departure to United.
Upon completing his move, Schneiderlin admitted that he cannot wait to begin linking up with Koeman on a daily basis in training again, and stressed that the Dutch boss will get the best out of his own game.
Schneiderlin told the Toffees' website: "Everton is a big club in the history of English football. I’ve always loved the atmosphere in the stadium. I’ve always liked playing here. I can’t wait to play and to represent this great club.
"There is a manager in place here that I know, I know he can get the best out of me, I know his style and how he likes to play football. He was very good with me from the start and I enjoyed playing football under him. At Southampton he gave me some great advice to keep improving every day and it was a pleasure to work with him.
"We spoke since the time I knew Everton would be interested. I know his ambition and why he came here. He believes in this and I do, too. We are trying to achieve something here and this is what I am here for. I know the fans have big expectations and we’ll do everything to make them come true.
"When I spoke with the manager and the people in place at Everton I understood immediately that they have a very good plan for the future of the football club. I just want to play football. I want to put my print on this club and do everything to get Everton where it belongs."
Koeman had identified central midfield as one of the areas that needed the most strengthening in January, and the capture of Schneiderlin will go a long way to instantly improving Everton's quality in the middle of the pitch.
The Frenchman could make his debut for the Toffees in the home clash with Manchester City on Sunday.