The Ivory Coast international spent seven years with the 20-time English champions, making 113 appearances, winning the EFL Cup, Community Shield and Europa League.
Bailly's most productive season came in 2016-17, when he made 38 appearances in all competitions, but he struggled with fitness problems after an impressive opening campaign.
The 29-year-old spent last season on loan at Marseille in Ligue 1, but the French club turned down the chance to sign him on a permanent basis.
Bailly did not leave Man United ahead of the English transfer deadline, but it was announced earlier this week that he had signed a one-year contract with Besiktas.
The Ivorian has said that he leaves Old Trafford with "many years of incredible memories", and he has thanked his teammates and the fans for their support.
"As I leave this club after many years of incredible memories, outstanding teammates, and trophies, I want to express my profound gratitude," he wrote on social media.
"This chapter may be closing, but the friendships and memories will forever remain in my heart. The fans here are the real champions. Thank you for everything, forever United."
In a statement earlier this week, Besiktas said: "Our club has reached an agreement with Manchester United for the final transfer of Ivorian defender Eric Bailly.
"We wish Eric Bailly, who we believe will provide important services to our club, great success with our glorious jersey, and present it to the public with our respect."
Man United are also still attempting to shift Donny van de Beek to Turkey before the division's transfer window closes next week, with Fenerbahce and Galatasaray both said to be interested.
However, Jadon Sancho is set to remain at Old Trafford, with a proposed loan move to Saudi Arabian outfit Al Ettifaq allegedly breaking down on Thursday due to the English club's demands over an obligation to buy.