He is now 10-under 209 for three rounds at the par-73 Plantation Course.
Meanwhile, Robert Garrigus (69) recovered from an atrocious start of a double bogey-bogey by holing a 50-foot eagle putt on the 18th hole to join Steve Stricker (65) and Jonathan Byrd (67) in the lead.
Facing the notorious Kona wind at Kapalua, Atwal began with a bogey like many other leading players. But he made up with birdies on third and sixth, before giving up another bogey on seventh.
But from there, the Orlando-based golfer, who became the first Indian winner on the PGA Tour last year, hit a purple patch. He birdied the eighth and ninth and then after turn added a third in a row at the 10th.
Continuing his fine run with the putter, he added two more on 13th and 17th to close at five-under 68.
Atwal has been steady with his driving and irons and for the past two days he has also had a good time with the putter.
He is currently the second best putter this week.
Garrigus went into the hazard with a 4-iron to open with double bogey and had a bogey on the next. But he fought back well and by the end, he was back where he started in shared lead.
The eagle gave Garrigus a four-under 69 and gave him a chance to become the first player since Tiger Woods in 2000 to win the season opener after winning the final event of the previous year.
Stricker had a run of five straight birdies that carried him to an 8-under 65, a score that looked tough in the wind.
Byrd, steady as usual, had a 67.
The three leaders, Robert Garrigus, Steve Stricker and Jonathan Byrd, all at 18-under 201, were three shots clear of Carl Pettersson, who had a 71.
No American has won the Tournament of Champions in the last eight years.
Matt Kuchar had the lead at one point by making seven birdies in a nine-hole stretch, but he played even-par over the final six holes for a 66 and was four shots off the lead.