The Braves have won three straight following a 1-5 stretch.
Toronto manager John Farrell, who called a team meeting before the game to let pitcher Ricky Romero address comments he made in criticizing the offense on Monday, watched the Blue Jays struggle at the plate for the third straight game.
Minor (1-2) allowed five hits, one run, one intentional walk and struck out a season-high eight in seven innings.
The Blue Jays have scored two runs in losing three straight for the first time since April 30-May 3. They have dropped eight of 12 overall.
Toronto led 1-0 in the first on Adam Lind's RBI single, but Chipper Jones' sacrifice fly tied it in the bottom half.
Atlanta took a 4-1 lead in the third on McCann's 12th homer and Alex Gonzalez's RBI double off rookie Zach Stewart (0-1), who allowed 10 hits, five runs, three walks and struck out two in 3 2-3 innings.
After Heyward put the Braves ahead 5-1 with an RBI single in the fourth, Stewart intentionally walked McCann to load the bases. Reliever Marc Rzepczynski limited the damage, though, by striking out Freddie Freeman on three pitches.
McCann, who went 2 for 3, is hitting .327 with 10 homers and 22 RBIs since May 17. The five-time All-Star catcher has reached base safely in 22 of his last 28 games.
Minor escaped a jam in the fourth after Lind singled and moved to third on Jose Molina's double with one out. But after striking out Rajai Davis and intentionally walking Mike McCoy, Minor held his 4-1 lead with a strikeout of Stewart. He retired the last 10 batters he faced.
Eric O'Flaherty and Jonny Venters combined to hold the Blue Jays without a baserunner in the eighth and ninth.
It was possible Minor, a first-round draft pick in 2009, would be optioned to Triple-A Gwinnett after the game with right-hander Brandon Beachy coming off the disabled list and starting Wednesday against Toronto.
The Braves have already called up Minor three times this season. Before Tuesday, the left-hander had gone 0-4 with a 9.07 ERA over his last nine starts.
Rzepczynski retired seven straight batters before Jason Frasor faced the minimum in the seventh. Octavio Dotel pitched the eighth.
Toronto slugger Jose Bautista, who began the game tied for the major-league lead with 21 homers, has gone deep just once in his last 22 games. He went 0 for 4 for the second straight night.
Stewart, whose ERA rose from 2.57 to 5.91, made his career debut last Thursday, receiving no decision in a 4-3 loss at Baltimore, allowing seven hits, two runs, one walk and striking out four in seven innings.
Notes: Jones returned to the lineup after missing the four games with a right adductor strain and went 1 for 3 with one run scored. ... Rzepczynski has held left-handed hitters to a .172 average (10 for 58) this season. ... Toronto has just one series win against the NL since 2004. ... Atlanta purchased the contract of veteran infielder Julio Lugo before the game and optioned Diory Hernandez to Gwinnett.