Dunn connected with one out against Tommy Hunter (3-2). The previous inning, Dunn misplayed a grounder by Matt Wieters, paving the way for the tying run to score and depriving Quintana of his first victory since May 21.
"The way Q pitched and how that inning went down, it was good any way we could to get a win," Dunn said.
Quintana struck out a career-high 11 in seven innings. He allowed two hits and retired his last 13 batters.
Reliever Addison Reed (2-0) wound up with the win.
Dunn's home run was his 23rd this year. He has batted .272 in June and July after a .156 average in April and May.
"I think that it's probably a little bit of everything," he said about his turnaround. "The things we've been working on since spring training just kind of paying off now."
"Like I said from Day 1, I've felt pretty good all year. Just wasn't getting any results. Now it seems like I'm swinging at a little better pitches and putting some good wood on it," he said.
Dunn also singled in a run to help Chicago take a 2-0 lead in the sixth.
"He's dangerous any time he goes up there. But again, just the quality of the at-bats is great to see," Chicago manager Robin Ventura said.
Quintana was locked in from the start.
"The beginning of the game I felt I had really good control and I felt I'm really going to be able to do some things," he said a translator. "That's what led to me striking out so many batters."
Quintana said the no-decision didn't bother him. His previous strikeout high was eight, set against Texas on July 5, 2012.
"Those are things you can't control," he said. "The most important thing to me is that the team wins. The best thing that I can do for the team is give the team a quality start, give the team the opportunity to win and give a quality outing."
Britton struck out three, walked one and gave up six hits over seven innings in his longest stint of the season.
"Both guys pitched real well," Orioles manager Buck Showalter said. "There was only going to be a small margin of error. We were fortunate to push over a couple there out of their bullpen."
"Quintana was tough. I mean, that was solid. I can see why they're so high on him. He's been pitching well for them. They've got a lot of good young pitchers but Zach matched him for the most part," he said.
Dunn decided the game by connecting on a curveball.
"It's a pitch you don't want to throw to him," Hunter said. "I would like to bury it. There were a couple righties coming up right after him."
Ventura said RHP Jake Peavy, on the disabled list since June 6, will throw Friday. Ventura said he doesn't expect Peavy to return from his broken rib until after the All-Star break. ... Baltimore signed South Korean LHP Jung-Hyun Yoon. He is slated to report to the Orioles' spring training complex in Sarasota to be evaluated for assignment within their system. ... The sixth inning has become a nemesis for Britton, who had given up a .643 batting average in sixth innings coming into the contest and then allowed two more runs. ... The starting time for the White Sox-Orioles game on Sept. 7 game in Baltimore has been moved to 1:05 p.m. from 7:05 p.m. Baltimore RHP Miguel Gonzalez (6-3, 3.77) starts Friday night at New York against Ivan Nova (2-2, 4.63) of the Yankees. The White Sox play at Tampa Bay Friday night, with Dylan Axelrod (3-4, 4.57) facing Jeremy Hellickson (7-3, 4.90).