The Braves trailed 7-0 after two innings but chipped away late and stopped a four-game losing streak.
Cleveland won the opener 8-4 behind Corey Kluber's strong outing.
Acuna ripped his double to left on the second pitch from Cody Anderson, Cleveland's third reliever in the ninth and a pitcher seeking his first big league save. The Braves also got a two-run single by Ozzie Albies and a bases-loaded walk to Freddie Freeman in the inning before Acuña, the budding star, came through after going hitless in his previous six at-bats.
Luke Jackson (2-0) pitched the eighth, and A.J. Minter struck out the side in the ninth for his second save.
Before Atlanta's comeback, the Indians had welcomed back All-Star Francisco Lindor, who finally made his season debut. Lindor returned after missing 19 games with an ankle injury, and went 1 for 5 with a run scored. More importantly, the talented shortstop was back atop Cleveland's lineup and his presence had an immediate effect.
The Indians scored seven runs in their first two at-bats against Touki Toussaint, whose first start this season couldn't have gone much worse. The right-hander gave up six hits, hit three batters and dropped the Braves into a deep, early hole.
Trevor Bauer kept them there for nearly seven innings.
He allowed only three hits - two by Tyler Flowers - in 6 1/3 innings and threw a season-high 120 pitches while making Atlanta's hitters guess what was coming next. Bauer has allowed four earned runs or fewer in his last 58 starts, the majors' longest streak since Hall of Famer Greg Maddux did it for 106 straight outings from 1991-94.
Dansby Swanson's two-run homer in the seventh got things for the Braves, who then sent 11 batters to the plate in the ninth.
Lindor could barely contain his excitement before the game, expressing his gratitude at playing again.
"I miss my teammates, I miss the game," he said. "I miss the things within the game. It's an honor. It's a privilege to be healthy. It's a blessing, and I ain't taking it for granted."
After receiving a boisterous ovation as he stretched while walking slowly to the plate, Lindor struck out in his first at-bat of 2019. An inning later, he singled sharply to right-center and scored on Ramirez's base hit. Lindor is wearing a brace on his ankle, which has slightly altered his gait but his boyish enthusiasm remains unchanged.
Jose Ramirez hit a two-run single and Jake Bauers added a two-run double during Cleveland's five-run outburst in the second, which was aided by Toussaint hitting Leonys Martin - for the second straight time - and nailing Carlos Gonzalez.
In the opener, Kluber bounced back from one of his worst outings in recent years by working seven innings. The right-hander didn't get out of the third in his previous start, when he walked a career-high five and gave up two homers in a no-decision.
Braves: Acuña stayed in the opener after being hit on the left elbow by Kluber in the third. Acuña also slid awkwardly while catching Allen's sinking liner in the same inning.
Indians: Injured starter Mike Clevinger has elected not to undergo surgery on an upper back strain, a procedure that could sideline him for three months. Francona said the right-hander will undergo another MRI and the results will determine when he can resume throwing.
Atlanta's LHP Max Fried (2-0, 0.92 ERA) will start the nationally televised three-game series finale against RHP Shane Bieber (2-0, 1.71 ERA), who has limited opponents to a .147 average in five appearances.
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