The 14-times major champion, playing in a high-profile group with four-times major winner Rory McIlroy (71) and reigning Player of the Year Justin Thomas (69), was forced to lean heavily on his short game given his erratic long game.
Defending champion and world number one Dustin Johnson battled to a 74 after an early triple bogey.
"Man, I fought hard, made a few mistakes, simple, silly mistakes, bad shots here and there ... but one-over's not bad," Woods told Golf Channel.
Woods struggled at times with a new driver that has a stiffer shaft than his previous version. He took comfort, however, that most of his wayward drives headed in the same direction, to the right.
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"The shaft is one I can go ahead and let it go," he said, explaining that he held back on a couple of drives.
"On a couple of tee shots I tried to hit little smoothies out there. The shaft is a little more stout so I can go ahead and hit it and when I did I piped it.
"The more I get used to it the more aggressive I can be. I can take out one side of the golf course, which is nice."
Woods made an ideal start with a birdie at his opening hole, the short par-four 10th, but quickly double-bogeyed the next after his drive clattered into a gum tree and disappeared, evidently stuck up in the branches.
Declaring a lost ball, he had to play again from the tee.
He finished the day having hit eight of 14 fairways, seven of 18 greens in regulation and needing 25 putts.
Woods will probably need to shoot another round of 72 on Friday to be sure of making the halfway cut.
He tied for 23rd three weeks ago at Torrey Pines in his first start of the year.
Leaders Finau and Cantlay posted five-under 66 on a day when a few players did not finish the round due to darkness.
(Reporting by Frank Pingue in Toronto and Andrew Both in Adelaide, South Australia; Editing by Christian Radnedge)
