Lester (9-7) picked up the win with his five-inning effort while the Cubs piled up 10 runs on his behalf through the first four innings.
But Atlanta rallied for eight runs over the last three innings and had the potential tying run at the plate in the ninth before Chicago finally secured a 14-12 victory.
"It was a little bit of a weird game for everyone," said Lester, who survived one rocky inning as he came off a 14-day disabled list stay for the start and win. (But) I felt like the ball was coming off pretty well. Maybe a couple of pitch selections to have back, but we won the game."
Rene Rivera clubbed a grand slam home run while Anthony Rizzo capped a 2-for-5 day with a bases-clearing triple as the Cubs ran their winning streak to six.
Braves starter Lucas Sims (2-5) dropped his second straight in three-plus inning outing, his shortest of the season.
"It was difficult, but we won it," said Cubs manager Joe Maddon. "We did a lot of good things and then we had to hang on. ... They (the Braves) really worked some good at-bats."
The Cubs (75-60) can claim their second consecutive series sweep with a Sunday victory. They took three straight from the Pirates earlier this week. The Braves (59-75) have lost three in a row.
Rivera clubbed his first career grand slam with one out in the second for 4-0 lead.
Acquired off waivers from the Mets on Aug. 19, Rivera launched Sims' 0-2 pitch down the left-field line where it ricocheted off the foul pole for his ninth homer of the season.
The Braves roughed up Lester for four hits and three runs in the third.
Freeman hit his 24th homer on a first pitch with one out to drive in Albies. Kemp followed with a first-pitch shot to left for his 15th homer and trimmed the Cubs' lead to 4-3.
Markakis then lined a double left but Suzuki grounded to third and Swanson flied to center as Jon Jay made a diving catch for the final out.
Baez made it 5-3 in the third with his 21st home, a solo shot to left on a 2-2 pitch with two out.
Lester gave up his third homer of the afternoon in the fourth on Rio Ruiz' shot to left as the Braves crept back to within one run at 5-4.
Sims walked Lester and gave up a base hit to Jay before being pulled in what turned into a five-run Cubs fifth. Left-handed reliever Ian Krol entered and walked Kyle Schwarber to load the bases for Rizzo's bases-clearing triple and 100th RBI of the season.
Rizzo trotted home on a two-out double by Baez and Jason Heyward singled to left to score Baez as the lead grew to 10-4.
Two of the runs were charged to Sims, who allowed seven runs on six hits while walking four and striking out four in his three-plus inning outing. Krol allowed three runs on three hits over two innings.
"You can't pitch like we did and expect to win," said Braves manager Brian Snitker. "It wasn't very good as we need. It was good that we came back and scored the runs, but we made way, way too many mistakes."
Lester was pulled for a pinch hitter in the fifth. He allowed four runs on eight hits, walked one and struck out three over a five-inning, 92-pitch outing.
In the sixth, Bryant scored from third on a Luke Jackson wild pitch for the Cubs' 11th run.
Atlanta cut the gap with three seventh inning runs, all charged to left-handed reliever Justin Wilson. The big blow was Markakis' two-run double scoring Albies and Freeman.
The Cubs got two runs back in their seventh on a two-run single by Albert Almora Jr. He brought home Rivera and Tommy La Stella to make it 13-7. Freeman's eighth inning double off Carl Edwards Jr. scored Lane Adams and Kemp's fielder's choice grounder plated Jace Peterson to cut the deficit to 13-9.
Heyward closed Cubs scoring with his ninth homer of the season in the eighth.
NOTES: Since reaching the .500 mark (45-45) with a win on July 16, the Braves have gone an NL-worst 14-30. ... Rookie INF Dansby Swanson has hit safely in 18 of his last 22 games. ... Braves LHP Max Fried (0-0, 5.40 ERA) makes his fifth appearance and first career start as he goes against Cubs LHP Mike Montgomery (5-6, 3.29 ERA) in Sunday's series finale. ... Chicago officially activated LHP Jon Lester off the disabled list prior to Saturday's start. They also called up newly-acquired OF Leonys Martin from Triple-A Iowa while designating LJHP Jack Leathersich for assignment.