Leishman had eight birdies and two bogeys at Firestone Country Club and was one stroke in front of England's Justin Rose, South African Charl Schwartzel and American Ryan Moore.
"I drove the ball well. It makes this course a lot easier when you're on the fairway," said Leishman, who had top-10 finishes at the US PGA Tour's National and the British Open.
"It's pretty tough when you're playing from the rough. Then, I hit my irons really well and was rolling the ball awesome. So it's a pretty good combination for a tough golf course like this."
Rose, the 2013 US Open champion who turned 34 on Wednesday, had five birdies in his five-under 65. Moore also notched five birdies without a bogey while former Masters champion Schwartzel had six birdies and one bogey for his share of second.
"I'm really happy with that number because it wasn't the most in-control round of golf," Rose said. "I kept the scorecard clean with a really good mix of some good iron play and good scrambling when I missed and that's what you have to do around here."
It was a further two strokes back to Americans Patrick Reed and Rickie Fowler, Canadian Graham DeLaet and Italy's Francesco Molinari on 67.
Former world number one Woods, an eight-time winner at Firestone, headed a group on 68 after a roller-coaster round featuring six birdies, two bogeys and a double bogey.
Woods, playing just his third event since back surgery on March 31, three-putted the first from eight feet.
He then hit his drive on the second so far left that it found the short rough to the right of the third hole.
The 14-time major winner recovered with a fairway wood onto the green and two-putted from 50 feet for birdie.
He bounced back from a bogey at the fourth with birdies at the fifth, sixth and eighth.
"Every time I dropped a shot I got right back on the next hole. I was three-for-three on that, which was nice," Woods said.
An errant drive cost Woods a double-bogey six on the ninth but he had two birdies with no bogeys coming in.
"I made my share of mistakes, but I hit a lot of good shots as well," Woods said. "I hit a lot of good shots on the back nine. The wind was blowing a little out there. The greens were soft, yes, you could be aggressive. But then again the rough was a little gnarly around the greens."
- McIlroy one-under -
Rory McIlroy, making his first start since claiming the third major title of his career at the British Open, headlined a group on one-under 69.
The 25-year-old teed off on 10 and had three birdies in his first 11 holes.
He three-putted the third hole -- his 12th -- for a bogey and made a double bogey six on the par-four eighth.
McIlroy responded with a closing birdie at the ninth, where he sank a two-footer to end on a good note.
"I didn't want any sort of let down and not just these two weeks, but basically the rest of the season," said McIlroy, who could have a chance to overtake Australian Adam Scott atop the world rankings with a win here.
Scott also opened with a 69 that featured three birdies and two bogeys, while reigning Masters champion Bubba Watson was also in the group of 14 players sharing 14th place at one-under.