The 33-year-old carded 69 to finish on 11-under-par 277 for the tournament, clinching his second PGA Tour victory and an invitation to the April 6-9 U.S. Masters.
Overnight leaders Charley Hoffman and Kevin Kisner, who both hit 73s, finished tied for second on 10-under.
Rory McIlroy, who will move up from third to second in the world rankings behind Dustin Johnson, needed a birdie at the last to draw level with Leishman but three-putted to finish two strokes behind after carding a 69.
"I holed some good putts on the last few holes. They all went in the middle of the hole, which is nice," Leishman said after a celebratory hug with his wife Audrey and their two young boys.
"Obviously the putter was there with me all week. The putter has been amazing this year. It's been really close and I just haven't played quite like how I wanted to on Sundays but today was the day."
Leishman blew away a strong field to win the Nedbank Challenge by six strokes on the European Tour in South Africa 15 months ago but this was his first PGA Tour victory since his maiden triumph at Hartford in 2012.
And it was Leishman's first victory in the U.S. since his wife nearly died of toxic shock two years ago. She is now healthy again and pregnant with their third child.
Leishman also took time to pay his respects to Palmer, who died in September at the age of 87.
"Mr Palmer was an awesome guy who I was lucky enough to meet a few times at this tournament," he said.
"Another reason why it’s so special this week, this is the first time I’ve won a tournament with my family here. It all came together.
Leishman, the third Australian to win at Bay Hill after Rod Pampling in 2006 and Jason Day last year, credited a new driver with his improved form.
"I’ve been playing well all year," he said.
"It’s really been a game changer. It’s funny, this game is a lot easier out of the fairway."
(Reporting by Andrew Both in Cary, North Carolina; Editing by Frank Pingue)
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