Atwal's card included four birdies, an eagle and three bogeys.
What it also got Atwal was a final round pairing with Woods (68) as the two are together at four-under 209. Woods and Atwal play with Rod Pampling in the final round.
Woods, playing for the first time in seven weeks, was steady for a three-under 68, but he was nine off the leader Briny Baird. Woods, who shot a second straight 68, had five birdies and two bogeys.
Briny Baird shot a seven-under 64 to go to 13-under 200 and become the outright 54-hole leader. It is the first time he is holding the lead in a PGA TOUR event in seven years and has not won in 357 starts in 12 years. Baird was two clear of Ernie Els (67) and Paul Casey (68).