Baghdatis, runner-up to Roger Federer at the 2006 Australian Open, won the first set before the players had a 75-minute wait in the dressing room during a rainstorm at Sydney Olympic Park.
Baghdatis fell behind 5-2 in the second set before breaking back twice for 5-5, then prevailing in the tiebreaker. It was Baghdatis' fourth career ATP singles title.
Baghdatis, who has won 16 of his last 17 matches, including a challenger event in Tashkent and the title at Stockholm, is unseeded at the Australian Open as he returns from wrist and back injuries last year.
He said he got an extra lift from winning a title ahead of the season's first major.
"Just won a tournament, your confidence is up," Baghdatis said. "Feeling pretty fit, playing good, moving good. So I think everything is positive."
Baghdatis was buoyed by a large pro-Cyprus crowd, including many of his relatives who live in the Sydney area.
"I want to thank my uncles, cousins, everyone, and it's my brother's birthday," Baghdatis said at the trophy presentation.
Baghdatis will play his first-round match at the Australian Open Tuesday against Italian Paolo Lorenzi, before a possible third-round clash with Lleyton Hewitt and a potential quarterfinal with Federer.
Gasquet played only his second tournament since a ruling by the Court of Arbitration for Sport allowed him to return to the court. A doping test last March showed traces of cocaine.
A panel of three lawyers from the CAS accepted Gasquet's claim that he inadvertently consumed cocaine by kissing a woman in a Miami club hours after withdrawing from a tournament with an injury.
The judges decided that the quantity of the drug in Gasquet's urine test was "minute" and that he was "clearly not a regular cocaine user."
Gasquet missed the French Open and Wimbledon while suspended, and in his first main-draw match after returning lost in straight sets to top-seeded Rafael Nadal at the U.S. Open.
He played in Brisbane last week.
"I lost to Marcos, a great player," he said. "For sure I feel confident for Australian Open, even if I have a tough draw."