Derek Lowe (8-11) allowed four hits and fanned seven batters in five-plus innings to win for just the second time in his last six starts. Arodys Vizcaino was charged with two runs over the next 1 1/3 innings. Eric O'Flaherty, Jonny Venters and Craig Kimbrel rounded out the contest on the mound.
Kimbrel pitched the final inning for his major league-leading 38th save, which is second-most all-time among rookies, two behind the record held by Texas' Neftali Feliz (2010).
The Braves have gone 18-1 since June 1 when O'Flaherty, Venters and Kimbrel have appeared in the same game. Kimbrel, who has not been scored upon in his last 29 2/3 innings, worked out of trouble in the ninth. Kelly Johnson bunted for a hit to lead off and Willie Bloomquist walked with one out. Collin Cowgill struck out and Justin Upton grounded out on a tapper in front of the plate to end the game.
Bourn and Jones both had two hits for the NL wild card leaders, who won for the fourth time in five contests.
"It was a team that we're battling for a playoff spot. We've had a tremendous last five games against teams that we're fighting for playoff spots for," said Lowe. "From that standpoint, it was gratifying that we went out and got a win."
Upton stroked a two-run double and Johnson had three hits for the NL West- leading Diamondbacks, who have lost three straight on the heels of a seven- game winning streak.
Daniel Hudson (12-9) surrendered seven hits and four runs over six innings in the opener of this three-game series.
Bourn led off the home first by cranking a 2-1 pitch over the wall in right field.
Lowe retired the final two batters of the third to escape a jam with a pair of runners in scoring position, and the Braves expanded their lead to 3-0 in the bottom half. Jose Constanza drew a two-out walk and McCann clobbered his 20th homer of the year, a shot to right field.
Upton was hit by a pitch and Miguel Montero walked to open the sixth before the Braves went to their bullpen. Vizcaino fanned Ryan Roberts and Lyle Overbay before getting Chris Young to fly out to keep it a shutout.
Jones then hit a ball off the top of the left field wall and out in the bottom of the sixth.
Upton doubled to the gap in right-center to bring home Sean Burroughs and Cowgill in the seventh, but Montero grounded out to cap the frame.
The Diamondbacks went down in order in the eighth and Kimbrel escaped his jam in the ninth.
Overbay struck out in all four of his at-bats...McCann has at least 20 homers in four straight seasons...Kimbrel easily leads all MLB relievers in strikeouts with 98...Venters has a 20 2/3 innings scoreless stretch...Jones has homered in back-to-back games...Arizona lefty Wade Miley will pitch Saturday in his MLB debut, while Brandon Beachy starts for Atlanta.