Manchester United have enjoyed a stellar three months with Solskjaer in temporary charge.
The Norwegian took over from Jose Mourinho in December and has inspired a top-four challenge in the Premier League.
United are also into the Champions League quarter-finals.
Solskjaer has been praised for his attack-minded style of football – something that was lacking at Old Trafford under Mourinho.
And the manager is determined to keep the fans entertained.
In his first message to supporters after signing a three-year deal, Solskjaer said: “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.”
The former United striker – who helped the Red Devils win the Champions League in 1999 – praised his staff for United’s excellent stint under his command.
“From the first day I arrived, I felt at home at this special club,” said Solskjaer.
“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.”
Manchester United executive vice-chairman Ed Woodward praised Solskjaer for the work.
“Since coming in as caretaker manager in December, the results Ole has delivered speak for themselves,” he said.