At 6-1 down on a tiebreaker, the Cypriot saved two match points before Tursunov sealed the victory with a backhand shot into the left corner of the court.
"I played rather simple but then started to create some knotty combinations and lost my service game," Tursunov said. "I then lost the initiative and he cheered up. I was lucky to outrun him on a tiebreaker."
Earlier, Guillermo Garcia-Lopez of Spain, who is also looking for his third career title, defeated Russian qualifier Mikhail Biryukov 6-4, 6-1.
Eighth-seeded Denis Istomin of Uzbekistan eliminated Samuel Groth 6-4, 3-6, 7-5, breaking decisively on his first match point when the Australian qualifier returned his forehand long.