The move has seemed inevitable for many weeks given the transformation United have undergone since Solskjaer arrived on a caretaker basis following the sacking of Jose Mourinho in December.
Solksjaer has won 14 of his 19 games in charge, reviving hopes of a top-four finish and unexpectedly guiding United to the quarter-finals of the Champions League.
Reacting to his appointment, the Norwegian told the club's official website: "From the first day I arrived, I felt at home at this special club. It was an honour to be a Manchester United player, and then to start my coaching career here.
"The last few months have been a fantastic experience and I want to thank all of the coaches, players and staff for the work we've done so far.
"This is the job that I always dreamed of doing and I'm beyond excited to have the chance to lead the club long-term and hopefully deliver the continued success that our amazing fans deserve."