Van de Beek, 23, becomes the club's first signing of the summer.
"I cannot begin to explain how incredible an opportunity it is to join a club with such an amazing history," Van de Beek said of the move. "I would like to thank everyone at Ajax. I grew up there and I will always have a special bond with the club. I am now ready to take the next step in my career and perform at the highest level and there is no higher standard than Manchester United."
"Donny has all of the technical attributes needed to perform in this team and has the personality required to succeed at Manchester United," Solskjaer said. "His ability to see space, time his movements and read the game will really complement the qualities that we have in midfield and his arrival really strengthens the depth of talent we have in that position.
"Donny's performances in the Eredivisie and in Europe have been excellent over the last few years and we are all looking forward to working with him."
Sources have told ESPN that United see a deal for Van de Beek as a good opportunity to strengthen the squad in a window during which they have struggled to agree a fee with Borussia Dortmund for Jadon Sancho.
Van de Beek had been linked with a move to Barcelona in recent weeks to link up with his former national team coach Ronald Koeman.
After coming through the academy at Ajax, Van de Beek made 175 appearances and won the league and cup double in the 2018-19 season.
He also played a key role in the club's run to the Champions League semifinals during that campaign where they eliminated European giants Madrid and Juventus.