Miller easily retired the first two batters in the ninth before Bour singled sharply on the first pitch, a 95-mph fastball.
"He put a good swing on it," Miller said. "I wish it would have finished differently."
Bour's hit came on the 89th pitch by Miller, who threw a strike on the first pitch to 26 of 29 batters and had 70 strikes among 94 pitches.
"You never want to be the last guy up there when someone gets a no-hitter," Bour said. "So I avoided that."
Braves catcher A.J. Pierzynski said the pitch to Bour was Miller's lone mistake.
"He wanted to throw it in off the plate, and he threw it right down the middle," Pierzynski said. "Those things happen. It stinks, but Shelby pitched a great game."
Miller walked one and benefited from an early replay reversal to flirt with what would have been the first no-hitter for the Braves in 21 years. The reversal erased an infield single by pitcher Henderson Alvarez in the third inning.
Alvarez nearly broke up Miller's bid in the third when he was ruled safe trying to beat out a soft grounder to short. Following a review that took three minutes and five seconds, the call was reversed to end the inning and erase the hit.
Replays on the video scoreboard appeared inconclusive and the crowd booed the reversal, but the Marlins didn't argue.
Sunday's game was the third time in Miller's career that he started and pitched the first five innings of a game without a hit, according to ESPN Stats & Information. Miller allowed a hit with two outs in the sixth inning in each of the two previous outings.
The Marlins' Giancarlo Stanton made the third out of the seventh inning by hitting a ball to deep left-center field where Todd Cunningham ran it down. The fly out was calculated at 389 feet and would have been a home run at four major league ballparks (Camden Yards, Wrigley Field, Miller Field and Citi Field), according to ESPN Stats & Information.
Miller, who struck out four, improved to 5-1 this season with a 1.33 ERA in eight starts.
"Obviously things didn't finish as we wanted to, but at the same time we got a sweep," Miller said. "In the end I had a lot of fun."
The Associated Press contributed to this report.