The second LIV Golf event of the season is at The Gallery north of Tucson, where the former Match Play Championship was held for two years.
For stroke play, the undulated greens and pins tucked on crowns made it difficult in spots to find birdies.
That wasn't an issue for Leishman. The Australian sunk 20 feet putts on the 14th and 16th before rolling in a 40-foot putt on No.3.
He also made a 12-footer for par on No.4. With only one bogey on his card, he led by one shot from Abraham Ancer, Louis Oosthuizen and Matthew Wolff.
Charles Howell III, who opened LIV's second season with a win at Mayakoba three weeks ago, had one of the best turnarounds.
He opened with a double bogey and was 3 over after four holes, only to answer with four straight birdies. He finished with a 69.
"I hung in there nicely," Howell said. "You're in altitude, the pins are on the edges of slopes, it's hard to get the ball close. Birdies aren't easy."
Howell is among eight players who previously played The Gallery when it was a World Golf Championship event in 2007 and 2008.
Henrik Stenson won in 2007 and lost in the semi-finals to Tiger Woods the following year before it moved to a different course on Dove Mountain.
Stroke play wasn't as kind to Stenson. He was 4 over after seven holes and had to rally with four birdies the rest of the way for a 73.
Dustin Johnson also got off to a rough start with a tee shot into the desert for double bogey on No.2, his starting hole in the shotgun start, and another double bogey on No.7. He finished with three straight birdies from short range for a 73.
Joaquin Niemann, Peter Uihlein and David Puig were at 67. The group at 68 included Sergio Garcia and Pat Perez.
Niemann and Puig each play for Torque, which had a one-shot lead in the team competition over Smash and Iron Heads.
Cameron Smith, at No.5 in the world the highest-ranked player with LIV Golf, struggled with his tee shots.
The Queenslander rallied with three straight birdies late in his round, only to hit another wild drive on the 18th hole for bogey and a 71.
Thirty of the 48 players in the field broke par. That included Phil Mickelson (70) and Laurie Canter (70), who again filled in for Martin Kaymer as the German recovers from injury.
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