The Norwegian has agreed to a deal until the end of next season, with an option to extend for another year. It is his first managerial role since being sacked by United in 2021.
Solskjaer took over as Manchester United's caretaker manager in December 2018, with the team sixth in the standings, replacing Jose Mourinho (now Fenerbahce manager) for the remainder of the season.
The Norwegian earned a three-year deal after a run of 10 wins out of 13 league games. He guided United to third and second-place Premier League finishes and a Europa League final.
However, a poor run of form led to his departure in November 2021 with United in seventh place in the league standings.
"I've been outside football for a little while, as a boss," Solskjaer told reporters.
"But I've been working in UEFA and I've been following football as well as this club... This project seems a little bit like Manchester United for me. We were sixth, then we became third and then second. And this is step-by-step.
"And of course, we're here to win, so hopefully we can make it faster. But we have to be realistic. And yes, I had many offers, but as I said before, it had to be the right offer worth me moving away from my perfect life."
Besiktas, who have won the Turkish Super Lig title 16 times, are currently fifth (at the time of writing) in the standings. They have been without a permanent head coach since Giovanni van Bronckhorst left in November.
"You need to feel that heartbeat racing again, because (you ask) 'do you have the energy?' and the project that the President and the board have been talking to me about, it was the right thing I needed now," he added.
"I've been following United (since I left) and it's difficult when it goes down. You need to just turn the boat and hopefully, we can turn the boat quickly here."
Solskjaer praised the club's young talent like Semih Kilicsoy (19) and Mustafa Hekimoglu (17). "A little bit unfair to compare with Erling Haaland," he added.
"Because Haaland is one of the best strikers in the world, but we will help them to become as good as they can get."
Solskjaer's side visit Mourinho's Fenerbahce in May in the Super Lig. Besiktas are 28th in the Europa League standings, a point off the playoff places, with two games left to play in the league phase.