Fighting a left wrist injury that will require surgery, Dunlap matched Quinn with a closing birdie on the par-5 18th on the Country Club of Virginia's James River Course.
"Maybe excruciating pain is the key to playing good golf because I'm not getting nervous on a shot -- you're just trying to get through it," Dunlap said. "The worst parts are gripping it and getting the club started ... that's when that bone hits that bone."
The top 72 players qualified for the Charles Schwab Cup Playoffs opener. The top 54 on Sunday will get spots next week in the Invesco QQQ Championship in Thousand Oaks, California, and the top 36 after that will advance to the season-ending Charles Schwab Cup Championship in Phoenix.
The 55-year-old Dunlap entered the week 29th in the standings. Playing through the wrist injury, he's coming off ties for ninth and seventh in his past two starts.
"I think I finally taped it the right way," Dunlap said. "Or maybe it's the pain meds kicking in. I don't know, one of the two."
Quinn is 64th in the standings.
"I finished up strong last year, too, kind of secured my privileges for the following year making eagle on 18," Quinn said. "I played solid all day. I had a lot of opportunities. A couple hiccups."
Defending champion Bernhard Langer opened with a 70. The 61-year-old German star won the SAS Championship last week in North Carolina to take the points lead. He has two victories this year and 38 overall on the 50-and-over tour.