Cole hadn't been on record for a loss since May 22, 2019, prior to the 5-1 setback Wednesday at Truist Park in Atlanta.
The Yankees also lost the second game of Wednesday's doubleheader and are on a five-game losing streak.
Cole allowed five hits and five runs in five innings to move to 4-1 on the season. His ERA jumped from 2.75 to 3.51 as a result of the performance.
His counterpart -- Braves starter Ian Anderson -- was dominant in his Major League Baseball debut, with one hit and one run allowed over six innings. Anderson also had six strikeouts and two walks. The only hit he surrendered was a sixth-inning homer from Yankees first baseman Luke Voit.
"It was pretty impressive," Braves manager Brian Snitker told reporters. "[Anderson] slowed [the game] down well. He had confidence in his pitches.
"He wasn't afraid of anything, I know that. He trusted his stuff. He got in the strike zone with those guys and that was really, really fun to watch."
Anderson got the Yankees out in order to start the game before Acuna stepped up to the plate in the Braves' first at-bat. The All-Star outfielder jumped ahead of Cole 3-1 in the count before he took a swinging strike on a fastball. Acuna then smashed a 3-2 Cole fastball over the left field fence for a 473-foot homer.
The solo blast traveled 114 mph and left the field in 5.8 seconds, according to Statcast.
Dansby Swanson pushed the Braves' lead to 3-0 with a 345-foot, two-run shot off Cole in the bottom of the third inning. Marcell Ozuna added a 469-foot solo blast in the same frame.
Anderson would carry a no-hitter into the sixth inning until Voit hit his 11th homer of the season for the Yankees' lone run.
Cole allowed a leadoff double to Freddie Freeman and walked Ozuna before he was taken out of the game in the bottom of the sixth frame. Yankees relief pitcher Luis Cessa then allowed an RBI double to Nick Markakis for the final run of the game.
Yankees left fielder Brett Gardner had his team's only other hit with a ninth-inning single off Braves relief pitcher Shane Greene.
Markakis went 2 for 3 with an RBI in the win. Swanson went 1 for 3 with two RBIs and a run scored for the Braves.