The 21-year-old Croatian will stay with the Bosnian club, which is owned by Vincent Tan, until the end of the season.
Delac joined Chelsea from Inter Zapresic in 2010 and has been on loan at a number of clubs since then, including Vitesse and Vojvodina.
"I came here after two or three months of negotiations," Delac told Sarajevo's official website.
"Now I am here and I want to help Sarajevo become the champion and also win the cup."
"This is a great day for Sarajevo and I think also a great day for Bosnia," club president Edis Kusturica added.
"For the first time in history a football club from Bosnia has signed a player from a Premier League team."