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.
Besiktas, who have won the Turkish Super Lig title 16 times, are currently sixth in the standings. They have been without a permanent head coach since Giovanni van Bronckhorst left in November.
"We believe Ole Gunnar Solskjaer will make significant contributions to our club, and we warmly welcome him to the Besiktas family, wishing him great success," Besiktas said in a statement.
The side host Samsunspor later on Saturday and travel to Antalyaspor next week 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.