Cameron Maybin homered leading off the 10th inning Sunday, the center fielder picking on a 1-2 fastball from reliever David Hernandez and sending his ninth home run of the season into the left field seats.
"I wish we could have got it for Shelby, but it feels great to get the win," said Maybin, who had never had a game-ending homer. "I wasn't thinking about going deep. I was just trying to put a good swing on it and get a hit."
Miller, who didn't allow a hit until the eighth inning, got a no-decision despite a two-hit effort with 10 strikeouts and now has gone 16 starts without a victory.
"At the end of the day, we got a victory and that's awesome," Miller said. "It was just trying to go out there and pitch. I know what that offense is capable of."
Arizona had averaged hit .354 and averaged 7.7 runs in their previous six games, but were baffled by Miller.
"Fanstastic, that's the only way you can describe him, and he didn't get the decision," Braves manager Fredi Gonzalez said. "He was nails. He walks the first two batters off the game and he was great after that."
Shortstop Nick Amed broke up Miller's no-hit bid with a bloop hit leading off the eighth and pinch hitter Jarrod Saltalamacchia followed with a hard single off the glove of second baseman Jace Peterson.
"I started to wonder there as the game went on how we were going to be able to muster up a run," Arizona manager Chip Hale said. "He had no-hit stuff."
Miller was pulled after 109 pitches and center fielder Ender Inciarte bounced into a fielder's choice against lefty reliever Ross Detwiler, tying the game.
Peter Moylan, pitching in the majors for the first time since 2013, retired the next two batters thanks to a running catch by Maybin in left-center field, but Miller had lost his chance for a victory.
David Aardsma (1-1) got credit after striking out the side in the 10th and watch Maybin win it in the bottom of the inning.
"He's been great," Gonzalez said. "He's had a heck of a year. ... He impacts the game defensively and offensively."
Miller, who has a 2.43 ERA, hasn't won since May 17, when he lost a no-hitter with two outs in the ninth inning at Miami and had to settle for a two-hit shutout.
The Braves had a chance to win it in the ninth, but Inciarte made a diving catch in center field with two runners on.
Maybin made Hernandez (0-3) a loser leading off the 10th, though.
The Braves (53-64) took the series, but have won just eight of their past 23 games.
Diamondbacks starter Rubby De La Rosa nearly matched Miller, working seven innings and allowed seven hits and one run. He walked four and struck out three.
Overthrowing early, Miller walked the first two Arizona batters before getting out to the first-inning jam thanks to strikeouts around a diving catch in foul territory by left fielder Michael Bourn.
Inciarte walked for the second time with one out in the third inning, but Miller kept the Diamondbacks hitless by blindly spearing Inciarte's one-hopper leading off the sixth with his back to the plate.
"I kind of got a little lucky on that play, obviously," Miller said.
Left fielder David Peralta drew Miller's fourth walk leading off the seventh inning, only to be stranded at first on two strikeouts and a comebacker to the pitcher.
It was the second time this season and fourth of his career that Miller had reached double figures in strikeouts. He also fanned 10 against Washington on June 30.
"The guy threw a great game against us and we battled to break up the no-hitter," Hale said. "We had our chances.
After wasting four hits and a pair of walks in the first four innings, the Braves finally scored against De La Rosa in the fifth.
Peterson had a leadoff single and shortstop Andrelton Simmons delivered a one-out double. Miller walked to load the bases and Bourn's bouncer was booted by Ahmed.
Arizona (57-59) has lost 13 of its past 16 games in Atlanta.
NOTES: The Braves promoted Moylan, 36, from Triple-A Gwinnett to add a veteran arm to the bullpen. He is coming off a second Tommy John elbow surgery and last pitched in the majors with the Los Angeles Dodgers in 2013. ... Rookie RHP Jake Brigham was optioned to Gwinnett by the Braves after failing to get an out in the ninth inning Saturday. ... RF Yasmany Tomas returned to the Arizona lineup after missing three games because of a calf strain, but was 0-for-4 with a strikeout. ... Braves 1B Freddie Freeman (oblique strain) began a short rehab stint at Triple-A. He is eligible to come off the disabled list Wednesday in San Diego, where the Braves begin a seven-game trip Monday. ... The Diamondbacks continue a 10-game trip with three games in Pittsburgh starting Monday.