Johnson birdied his final hole in regulation to get into the play-off then birdied the par-four 18th with a putt from off the green.
South Africa's Grace had putted first in the play-off, also from the fringe, narrowly missing.
Australia's British Open champion Smith was closest to the hole, but his putt was never on track.
Johnson scooped the $4 million first prize from the $25 million purse and joined Talor Gooch - winner of the previous two events - and Brooks Koepka as two-time winners in the Saudi-backed breakaway circuit's first 14 events.
The former world number one had claimed his first LIV title at Boston last year.
Atop the leaderboard going into the third and final round, Johnson shook off a triple-bogey on the 10th hole to card a three-under 67 to join Grace and Smith on 17-under.
"Obviously after I made seven on 10 - everything I could do wrong I did wrong on the hole -but to fight back and birdie 18 two times in a row, I'm really happy with that," Johnson said.
Smith, five back with nine to play, carded a bogey-free nine-under-par 61 while Grace - who shot 61 in the first round - closed with a five-under 65.
Grace helped South Africa's Stinger team, headed by Louis Oosthuizen, to capture the team title with a 40-under-par total.
The next LIV tournament is at Trump National in Washington in two weeks. But next week Johnson and Smith will be among 18 LIV rebels teeing it up in the second major of the season, the PGA Championship at Oak Hill in Rochester, New York.
LIV, bankrolled by Saudi Arabia's Public Investment Fund, split the sport last year by luring away top stars from the US PGA Tour and the DP World Tour with lucrative contracts.
Amid the acrimony, organizers of the major championships have maintained qualifying standards that allow LIV golfers to compete, setting the stage for them to take on loyalists including Masters champion Jon Rahm of Spain and world number two Scottie Scheffler at Oak Hill.
"I'm really playing well," Johnson said. "Got a lot of momentum going into next week."
