Fognini, the runner-up last year, broke Inglot early in each set and saved the only break point he faced late in the second set.
The 17th-ranked Italian is looking for his third career title. He won twice in June.
The 34th-ranked Russian was serving for the match at 5-2 and 5-4 in the second set, but Baghdatis broke him twice to force a tiebreaker.
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."