The 51-year-old Norwegian, who was axed by United in November 2021, takes over from Dutch coach Giovanni van Bronckhorst who was sacked by Besiktas at the end of November.
"A contract has been signed with Ole Gunnar Solskjaer... until the end of the 2025/26 season, with the 2026/27 season as an option," said the Istanbul-based side in a press release.
Solskjaer made over 200 appearances as a striker for United and was a member of their treble-winning side in 1998/99, scoring the winner against Bayern Munich in the Champions League final.
After his playing career, he went into coaching, spending a long period with Norwegian side Molde before rejoining United in 2018.
He became a technical observer for UEFA after losing his job at Old Trafford.
Besiktas are currently sixth in the Turkish Super Lig, 21 points behind leaders Galatasaray and 12 behind second-placed Fenerbahce, coached by Jose Mourinho, whom Solskjaer succeeded in Manchester.