Miami was shut out for an MLB-high 14th time, and its losing streak is now at six.
Atlanta displayed its firepower rather quick, pounding out two homers and four runs in the first inning. McCann kept the pressure on Turner, connecting in the third inning.
"I don't think anybody is trying to do a little more. Maybe a few guys aren't clicking, pitchers and hitters, both," Turner said. "When we were playing real well, it seems like everybody was kind of clicking and doing what they're supposed to do every night.
"For myself personally, things aren't clicking as smoothly as I'd like. It's something we've got to work on and get back to where we are clicking."
For McCann, his solo shot was the 21st of his career against Miami, his most against any club. The All-Star catcher has 20 against the Phillies.
Turner entered the game having allowed three home runs in his first 12 starts, a span of 77 1/3 innings. Ignoring the statistic, the Braves matched that total in three innings.
Given early support, Brandon Beachy coasted on his way to his first win of the season, tossing eight scoreless innings of three-hit ball. The right-hander celebrated his first victory since returning from Tommy John surgery.
"That's 14 in a row," Beachy said. "That's fun. Hopefully we can keep that going. It makes pitching a little bit easier when those guys work their tails off and put five up for me. I don't have to be fine. I can go right at guys. I tip my cap to them."
Yelich's single to start the game extended the rookie's hitting streak to nine games.
The slumping Marlins have fallen to 43-71. Being 28 games under .500 matches their low point of the season. They were 13-41 on May 30.
"These guys have been playing great, obviously," manager Mike Redmond said. "They came out swinging the bats. Jacob made a couple of mistakes out over the plate, and they made him pay. They put a four-spot up in the first, and we couldn't answer."
Turner's night was done after five innings, as he allowed five runs (four earned) on 96 pitches.
"Turner made some mistakes over the plate, and we made him pay for it, and I think that gave Beachy a chance to kind of catch his breath a little bit and cruise on," Braves manager Fredi Gonzalez said. "We added one more later on, and he made it stand."
There was plenty of power and strong emotions in the Braves' four-run first. Upton homered with one out to trigger the rally. Freddie Freeman singled to right, and McCann reached on shortstop Adeiny Hechavarria's error.
The mistake created an opening for a team that already is putting up plenty of runs. Johnson connected on a three-run homer, which even further energized the tomahawk-chopping crowd.
"I just didn't make pitches," Turner said. "The pitch to Upton was just a fastball right down the middle, and he's obviously swinging a hot bat. Then, the pitch to Chris Johnson was just a really bad slider, right down the middle. He's leading the league in hitting. Two guys who are obviously, really good hitters."
Two pitches later, Turner's 90-mph fastball ran up and in on Dan Uggla, plunking the second baseman on the left arm. Taking exception, Uggla exchanged a few words with Turner. Catcher Jeff Mathis and home-plate umpire Mike Estabrook shielded Uggla from Turner, who was making his case that the pitch wasn't intentional.
Pitching coach Chuck Hernandez visited the mound to settle Turner, and warnings were issued to both benches.
"I just didn't have great command of my fastball," Turner said. "I'd never try to throw at somebody's head like that. That's what I told him. I know that it might have looked bad, giving up a couple of homers. But it was not intentional, by any means."
As the inning ended on Beachy's fly ball to right field, Uggla crossed paths with Turner, and gave the pitcher a friendly pat.
"I think it was just one of those heat-of-the-moment things," Turner said. "I can understand, as a batter, you never want to have a pitch up there like that. I was just trying to find my command."
For the Marlins, it was much of the same against the Braves, who hold a 7-3 season series edge.
"You run into a team that has been playing well," Mathis said. "We left a few balls out over the plate, and they did to it what a good lineup can do. They put some runs up on the board."