Batting right-handed, Albies lofted a towering two-run shot to right-center field off left-hander Mike Montgomery (1-2) in the seventh for his 12th homer to put Atlanta ahead.
The Braves held on to win the second of a four-game series between the NL division leaders.
Albies is hitting .550 (11 for 20) against the Cubs this season, with two homers. In his third season, the 22-year-old from Curacao is clicking at .483 (29 for 60) with four homers and 11 RBIs against Chicago in his career.
''He's very good,'' said Cubs manager Joe Maddon, who was aware of Albies' numbers. ''Historically (Montgomery) puts the ball on the ground, but it was just a high changeup.''
Fried (9-3) got the win despite a season-high five walks. He struck out eight and allowed two runs in winning his second straight start - and improving to 3-0 with a 1.59 ERA in three career starts against the Cubs.
The 21-year-old Acuna crushed the first pitch of the game from Cubs rookie Adbert Alzolay to the back of the left-field bleachers for his 19th homer.
That's all the offense Atlanta needed to improve to 17-6 in June.
''I think, for us, it's fun to be able to go out and try and help the team every day,'' Acuna said.
Albies and Acuna have found a spot in the ascending Braves' lineup.
''I feel like they've been here forever, but you know they're right along those same lines as Max,'' Braves manager Brian Snitker said. ''They're still learning things. . two very talented young players and they get on-the-job training every night they go out there.''
Relievers Sean Newcomb, Anthony Swarzak and Luke Jackson, who threw a perfect ninth for his 12th save, combined to nail down a four-hitter. Newcomb pitched one-plus inning in his first game since being hit in the head by a line drive 10 days ago.
Willson Contreras had two RBIs and fellow Venezuelan Alzolay tossed one-run, one-hit ball through 4 2/3 innings in his first major league start.
After Acuna launched Alzolay's fastball just short of the left-field video board for a quick 1-0 Atlanta lead, the 24-year-old right-hander retired 12 of the next 13 Braves batters. He was replaced by Montgomery after walking the bases loaded in the fifth and the Cubs leading 2-1.
''He was outstanding,'' Maddon said. ''I think he just kind of hit a wall there at the end.''
Alzolay tossed four innings of one-hit relief in his major league debut last Thursday and got the win.
Fried walked three in the second to load the bases with one out, but dodgy Cubs baserunning led to a rally-killing double play.
Catcher Brian McCann caught Javier Baez flatfooted about 25 feet down the third base line and fired to Josh Donaldson who tagged Baez for the second out. Donaldson threw to Dansby Swanson who tagged out Contreras as he tried to advance to third.
Contreras made amends in the fourth with a double down the left field line that extended his hitting streak to nine games and put Chicago ahead 2-1.
Charlie Culberson made a sliding catch in left center to rob Baez of a hit in the ninth.
Closer Craig Kimbrel was slated to pitch at Triple-A Iowa on Tuesday night, then Cubs management planned to evaluate his progress. The right-hander, who finalized a three-year $43 million deal with the Cubs on June 7, could be activated this week.
Braves: Returned LHP Newcomb from his rehab assignment and reinstated him from the seven-day concussion IL. RHP Chad Sobotka was optioned to Triple-A Gwinnett to make room on the roster. Newcomb was hit in the back of the head by J.T. Realmuto's liner in start against Philadelphia on June 15. ... Rookie RHP Mike Soroka (right forearm) played catch on Tuesday after being hit by a pitch in Washington on Sunday. Manager Brian Snitker said Soroka could throw off the mound on Wednesday.
LHP Dallas Keuchel (0-1, 5.40) makes his second start for the Braves and faces Chicago RHP Yu Darvish (2-3, 4.75) on Wednesday night. Keuchel, who signed a one-year, $13 million contract with Atlanta on June 7, labored through five innings and lost in his Braves debut at Washington last Friday. Darvish has no-decisions in his last 10 starts and has yet to win at Wrigley Field as a Cub.