Leishman finished the tournament at 14-under 266, a stroke in front of 2010 Travelers winner Bubba Watson and Charley Hoffman, who was left ruing a double-bogey, bogey finish that dropped him from 16-under.
"It's an incredible feeling, I didn't think I would be in this position when I started this morning," Leishman said.
"This isn't the way I thought I would pick up my first win, but I'm not complaining. I'll take a win any way I can get it."
Roland Thatcher also had a disappointing finish when, needing a birdie at the last to force a playoff, he hit his approach from the fairway into a greenside bunker and finished with a bogey.
Thatcher finished at 12-under in a tie for fourth with Briton Brian Davis, South African Tim Clark and fellow American John Rollins.
Sweden's defending champion Fredrik Jacobson finished with a 68 to tie for eighth at 11-under.
Leishman went out early and posted a good number that turned out to be enough for a breakthrough win.
Starting a good two hours before the final group, the 28-year-old Victorian wasted no time getting to work and carded five birdies on the front nine.
Leishman credited a bad finish to his third round 70 for providing him with motivation for Sunday's blistering round.
"Yesterday I doubled 17 and bogeyed 18 and it fired me up to really go out and do some damage today," he said.
Fire him up it did as he went out in 30 and then added three more birdies coming in to cap off a flawless round.
Leishman then faced a long wait as his various pursuers found a number of different ways to throw their chances of winning away on the closing holes.
"The wait was the hardest part, you don't really get too nervous out on the course," Leishman said. "I got something to eat, watched some soccer and then put the golf on. Then I hit some balls and putted.
"I'm glad that turned out to be a practice session and not a warm up session."
There was no playoff required, though, because Hoffman was the only player to match or pass Leishman's 14-under.
Hoffman got to 16-under after a birdie at the 13th hole but a pushed tee shot into the water at the penultimate hole led to a double-bogey and he closed with a bogey after failing to get up-and-down from a greenside bunker.
"Just got a little quick on 17, it's a tough tee shot for me," Hoffman said. "What I did on 18 was pretty pathetic. Fanned the drive out to the right, pretty poor second shot, pretty poor bunker shot and even worse finish.
"All said and done, a bad finish and a bad taste in my mouth, but you learn from it."
Leishman was still not out of the woods as Thatcher, who entered the day tied for the lead with Davis, came to the 18th needing to birdie to get to 14-under.
To the Australian's delight, Thatcher's approach shot landed in a bunker and Leishman was finally able to celebrate his first tour title.