Arizona will send a three-player package highlighted by shortstop prospect Dansby Swanson, the No. 1 overall pick in this year's draft, to Atlanta in exchange for Miller. The Braves also will receive outfielder Ender Inciarte and pitching prospect Aaron Blair, while the Diamondbacks will also receive minor league pitcher Gabe Speier.
The Braves had previously insisted that Arizona includeA.J. Pollock in any potential deal for Miller, but the star outfielder was not included in the trade, which was agreed to Tuesday night.
Sources told ESPN earlier Tuesday that the Diamondbacks were closing in on a deal with free-agent right-hander Mike Leake. But a source told ESPN's Jerry Crasnick that Arizona's interest in Leake effectively ended with the Diamondbacks' acquisition of Miller.
Miller, who went 6-17 last season despite posting a career-best 3.02 ERA, will join Cy Young runner-up Zack Greinke atop Arizona's rotation. The Diamondbacks and Greinke finalized a six-year deal earlier Tuesday that, according to sources, is worth $206 million.
Miller, 25, spent one season with the Braves after playing for the St. Louis Cardinals from 2012 to 2014. He had a 24-start winless streak in 2015 but also posted career bests in innings pitched (205) and strikeouts (171).
"This was another trade that will brighten our future and give us impact-type players not only tomorrow, but right now," Braves president John Hart said in a statement. "We feel each of the guys we got back has impact-level talent along with a tremendous make-up."
Swanson, 21, was selected with the top overall pick June 8 after leading Vanderbilt to the College World Series, in which the Commodores lost to Virginia in the championship. Swanson played 22 games at Class A Hillsboro and batted .289 with one home run in 83 at-bats. He becomes the fourth No. 1 overall pick in MLB history to be traded by his drafting team before reaching the majors.
Inciarte, 24, batted .303 last season and could have a spot as Atlanta's leadoff hitter. He had six homers, 21 stolen bases and 73 runs scored in 132 games in 2015.
Blair, 23, went 13-5 with a 2.92 ERA last season in 26 combined games with Double-A Mobile and Triple-A Reno.
Atlanta got Speier from Detroit in the trade for Cameron Maybin last month. The 20-year-old Speier has a 2.34 ERA in 45 games -- eight starts -- over three minor league season. He made 33 appearances for Class-A West Michigan this season.
ESPN's Stats & Information contributed to this report.