Atwal fired three birdies on the ninth, tenth and 18th holes which were negated by bogeys at the fifth, 11th and 15th holes.
Woods, playing after a long gap, had an early birdie and then went 13 holes without his second birdie.
Valero Texas Open winner Brendan Steele shot four-under 67 at CordeValle and he was joined in the lead by Briny Baird, Garrett Willis and Matt Bettencourt.
Bhasin makes cut
Kunal Bhasin gave himself a good chance to register his best finish of the season so far after he added a second round 73 to make the cut and position himself at 26th shot at the midway stage of the Yeangder Tournament Players Championship in Taipei on Friday.
Bhasin, who had a 72 on the first day, is now one-over 145 for 36 holes but only six behind the leader -- Malaysia’s Khor Kheng Hwai (68) and Chinese Taipei’s Chan Shih-chang (67) -- both of whom are at five-under 139.
Poor start by Kapur
Shiv Kapur got off to a poor start with a four-over 76 to be placed tied 114th after the opening round of the Bankia Madrid Masters.
The Indian had two each of birdies, bogeys and double bogeys in his round on the first day. And now he will need a very low score in the second round to make the cut.
Meanwhile, Ross McGowan (64) returned to form in sensational style to lead the Madrid Masters after the opening round.