The 29-year-old midfielder, who joined Metalist from Napoli in 2011, spent last season on loan at Atletico Madrid, winning the Primera Division title with Diego Simeone’s side.
And, on Monday afternoon, Besiktas confirmed his arrival in a statement on their official website.
"I was told good things about Besiktas and their fans," Sosa said. "I am very happy to be here."
The Turkish giants will pay a €1.4 million (£1.1m) loan fee, with the option to make the transfer permanent for an additional €2m (£1.6m).
Sosa was a late addition to the Argentina squad at World Cup 2014 and, following injuries to the likes of Angel Di Maria, became a regular in the side who eventually ended runners-up after going down 1-0 to Germany in the final.
He joins fellow recent arrivals Demba Ba, Ramon Motta and Gokhan Tore as the club prepare to challenge for what would be their first league title since 2009.
Slaven Bilic's side opened their Turkish Super Lig campaign with a 1-0 victory over Mersin Idmanyurdu.