The former world No.1 started Friday just one shot off the lead.
Once on the green, a three-putt resulted in a triple-bogey seven but he stormed home with a birdie on the ninth and five more on the back nine. It was his first ace on tour.
He'd reeled off an eagle and four birdies to sit one shot off the lead at the winners-only event, prior to giving an interview near the 16th green at Kapalua's Plantation course on Maui.
Defending champion Johnson kicked off his bid for a third Hawaii title by mixing seven birdies with a sole bogey, while Thomas moved up the leaderboard with four birdies in his last five holes.
Tway qualified for this winners-only field by capturing the Safeway Open for his first PGA Tour victory.
"I stayed patient out there and it was good also to play a couple of practice rounds in conditions like what we had today".
Johnson accidentally hit the wrong ball from the native area following an errant drive, and was assessed a two-shot penalty which resulted in a double bogey.
World number eight Rory McIlroy and No. 14 Jason Day are tied for sixth at four under, while world number six Jon Rahm is tied for 11th at three under.
"I will need to because Gary is playing great golf, then Rory is just behind him and, as big hitters, they are likely to make a lot of birdies".
"It has been a nice start to the year", McIlroy said.
The Tournament of Champions is an elite $US6.5 million ($A9.3 million) event for winners on the PGA Tour from the previous year and doesn't have a 36-hole cut.