Solskjaer arrived as United's caretaker manager in December and has since lost only three of his 19 games in charge, steering his old club back into top-four contention and into the Champions League quarter-finals.
United won 10 of his first 11 matches in charge before finally succumbing to defeat against PSG, but the Red Devils overturned a 2-0 first-leg home defeat to make history and beat the Ligue 1 champions 3-1 in Paris.
In an official statement, executive vice chairman Ed Woodward said: “Since coming in as caretaker manager in December, the results Ole has delivered speak for themselves.
“More than just performances and results, Ole brings a wealth of experience, both as a player and as a coach, coupled with a desire to give young players their chance and a deep understanding of the culture of the club. This all means that he is the right person to take Manchester United forward.
“I want to thank Ole and the coaching team for everything they have done so far and congratulate him on this richly deserved appointment. The fans and everyone at the club are behind him as he looks to take us where we need to be and build the next stage of our history.”
Solskjaer added: “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.”
Solskjaer replaced Jose Mourinho as a surprise caretaker appointment in mid-December, initially joining 'on loan' from Molde until the end of the season.
It was reported that a clase in his contract allowed United to take him from the Norwegian side – the only club allowed to do so – where he had a contract through to 2021.
But after a significant upturn, United have finally tied down the 46-year-old, who spent 11 years at Old Trafford as a player and made 366 appearances.
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