Woods made the cut with a shot to spare after a disappointing second round of 72, although all 79 players who qualified for the last 36 holes were separated by just nine shots.
That meant the 15-time major winner had characteristically not given up hope of adding to his tally at Harding Park, where he won the WGC-American Express Championship in 2005 and compiled a perfect 5-0 record in the 2009 Presidents Cup.
However, Woods got off to a sluggish start on Saturday with seven straight pars before three-putting the eighth from 30 feet to fall back to one over par, nine shots behind halfway leader Li Haotong of China.
World number one Justin Thomas had earlier given hope to the chasing pack in the early stages of his third round in pursuit of a second US PGA title.
Thomas, who lifted the Wanamaker Trophy at Quail Hollow in 2017, made the cut on the mark of one over par, but closed to within four of the lead after following a birdie on the second with four in a row from the fourth.
A bogey on the eighth halted Thomas' charge and another on the ninth dropped him back to two under, although the 27-year-old stopped the rot with a birdie on the par-five 10th.
Two-time US PGA winner Rory McIlroy, who played alongside Woods and Thomas for the first two days, began his third round with two pars to remain one under par.
The final pair of Li and Fleetwood were due to begin their third round at 1500 local time (2300BST).