The Red Devils paid a fee of £30.7m for the Swede, with add-ons worth up to £8.8m also included in the deal.
The 22-year signed has signed a four-year contract, with an option for a further year, which will come into effect on July 1.
Manager Jose Mourinho, who failed with a bid for Lindelof in January, was pleased to complete his first deal of the summer.
"Victor is a very talented young player, who has a great future ahead of him at United," said Mourinho.
"Our season last year showed us that we need options and quality to add depth to the squad and Victor is the first to join us this summer.
.@VLindelof: "I'm thrilled to join United. I'm keen to get started and contribute to the team's efforts to win more trophies." #HejVictorpic.twitter.com/FoC6IuUWwj
— Manchester United (@ManUtd) June 14, 2017