The 22-year old full-back jets out of Melbourne on Tuesday night and will be eligible for selection for Bursaspor's Turkish SuperLig clash with Galatasaray this weekend subject to international clearance.
"To make a move to a club that plays Champions League football is a dream come true and I really can't thank Melbourne Heart enough for giving me this opportunity," Behich said.
"I have loved my time at Melbourne Heart and all the support that I have received from the members since I joined.
"I will miss them and miss being around my old team-mates."
Heart head coach John Aloisi praised Behich for the improvement he has made in the past few years and is proud of the fact he gets to compete with the likes of Didier Drogba and Wesley Sneijder who recently joined Galatasaray.
"It hurts to lose him at this stage of the season but how can you stand in the way of a player fulfilling his dreams.
"It's a great opportunity for the club to continue to grow and a chance for Aziz to establish himself as a genuine international for Australia."