Leishman, who entered the day six shots off the pace, charged up the leaderboard and then had to sweat it out in the clubhouse as all his pursuers tried to best him down the stretch.
But in the end, no one matched the 28-year-old Australian's 14-under 266, giving him his first crown.
Masters winner Bubba Watson (65) was within striking distance at minus-13 through 13 holes, but that was all the closer he would get. He finished one stroke behind.
Charley Hoffman (66) also posed a challenge, but carded a double bogey at 17 to end his threat, finishing tied with Watson for second at 13-under 267.
Third-round leaders Brian Davis (70) and Roland Thatcher (70) also saw their bids come up short as each fell to bogeys on the back nine.
Thatcher failed to birdie the final hole, handing the title to Leishman.