Alvarez kept Atlanta in check, and catcher Jarrod Saltalamacchia delivered the crucial hit with two outs in the ninth inning off one of the toughest closers in the game.
Steve Cishek locked down the bottom of the ninth for his third save in the series, and fourth in five days. Miami was able to bounce back after losing, 6-1, on Wednesday to claim three of four at Turner Field.
"This has been a really tough place for us to play," Miami manager Mike Redmond said. "It's good to see us find a way. It wasn't perfect, by any means. Henderson did a great job."
With the score deadlocked at 2, Saltalamacchia ripped an RBI single to center off Craig Kimbrel. A pair of wild pitches by the Atlanta closer, who was in the game in a non-save situation, proved costly.
Marcell Ozuna struck out in the ninth, but he reached first safely on a Kimbrel wild pitch. The closer's second wild pitch of the inning advanced Ozuna to second, and the Miami center fielder scored on Saltalamacchia's single to center.
"Fish-style, I guess," Redmond said. "Strike three, two wild pitches and a knock. We'll take it."
Saltalamacchia had a big night, going 3-for-4 with a run scored and the game-winning RBI. He also threw B.J. Upton out trying to steal second in the third inning.
Saltalamacchia's single off Kimbrel came on a slider.
"He's an All-Star closer. He's got great stuff," Saltalamacchia said. "Really, he has a great idea of what he's doing out there. In that situation, I'm just trying to not get too big and put the ball in play. He threw me a lot of sliders tonight, which wasn't what he's done in the past. But I struck out two nights before on a slider, so I can't blame him. He made a mistake I was able to do something with."
Miami took three of four for the first time at Turner Field since June 4-7, 2007, while Alvarez beat Atlanta for the first time in his career. He entered the night 0-2 with a 6.67 ERA in five career starts against his division rival.
"This year since the season started, I've gone out to work and give 100 percent for my team and battle," Alvarez said. "Every time I go out on the mound, I'm focused on doing my job."
Alvarez was impressive in his first outing since he sustained a left shin contusion. In his previous start, on July 19 against the Giants, the All-Star was struck just above the ankle on a comebacker. The bruise caused him to leave the game after 2 2/3 innings.
Miami had chances off Aaron Harang, who gave up two runs on seven hits with five strikeouts in seven innings. But he helped his cause by getting double plays in the fourth and sixth innings. And in the seventh, after Saltalamacchia's leadoff double, the veteran stranded the Miami catcher at second.
For the second straight game, the Marlins were victimized by a second-inning, two-out Atlanta home run. On Wednesday, Freddie Freeman did the damage with a three-run shot off Nathan Eovaldi.
In the series finale, it was Chris Johnson who went deep, connecting on a two-run homer off Alvarez to give the Braves the early advantage.
"He's tough," Johnson said of the Miami right-hander. "You've got to try to play small ball and take advantage of situations that he gives you. Tonight, he didn't give us too many."
The Marlins pushed home a pair of runs in the fifth inning, tying the score. Harang put himself in a bind by walking Ozuna to open the inning. Saltalamacchia doubled, and Adeiny Hechavarria floated a soft liner to right for an RBI single. Christian Yelich's fielder's-choice grounder to second drove in Miami's second run.
"These division games mean everything," Saltalamacchia said. "That's when you can gain some ground. To be able to take three of four is big for us."