Leishman, who was six shots off the lead after 54 holes and tied for 20th place, had a bogey-free round with eight birdies. He finished at 14-under 266.
It was his second career victory on the PGA Tour, to go with the 2008 WNB Colf Classic.
Masters Tournament champion Bubba Watson (65 on Sunday), and Charley Hoffman (66) tied for second place.
Hoffman had control of the tournament until he made double bogey on No. 17 and double on No. 18.