Adam Duvall belted a homer over the Green Monster and Nick Markakis had three hits and drove in two runs for Atlanta, which entered the three-game set winless (0-12-3) in its last 15 series against the Red Sox.
"That's the goal every time I go out there," said Fried after the Braves improved to 8-0 in his starts this season. "I didn't have the best stuff and put us down."
After trading five players off its MLB roster in the last two weeks, Boston, buried in last in the AL East, lost for the 23rd time in 35 games in this pandemic-shortened season.
Alex Verdugo had three doubles for the Red Sox.
Fried gave up two runs for just the second time in eight starts this season, breaking his string of six straight allowing one or none. In five innings, he gave up five hits, fanned five and walked two.
"He wasn't real sharp," Braves manager Brian Snitker said. "That's the thing about Max, he keeps making pitches and got through it."
Mark Melancon got the final three outs for his seventh save in eight chances.
With the score tied at 2-2 in the fifth, the Braves loaded the bases against starter Colten Brewer (0-3) before Riley's triple off the top of a low wall down the right-field line on the second pitch from reliever Phillips Valdez.
"I forgot how low it was down there," Riley said. "I thought it was foul at first. I was lucky to get a triple."
Duvall's homer left Fenway Park and tied it in the fourth after Fried balked home a run.
Brewer was charged with five runs in four-plus innings.
"We tried to give him a chance to get a win and try to get him actually into the fifth inning. He's been going three or four. Thought because his pitch count was down we'd do it, and it didn't work so well," Boston manager Ron Roenicke said.
After getting into the area around 4 a.m. following their wild, four-hour plus victory over the Phillies in Philadelphia on Sunday night, the Braves jumped ahead 1-0 on Markakis' RBI double in the first.
Rafael Devers' run-scoring single tied it in the bottom half.
Braves: OF Ronald Acuna Jr. was out of the lineup with tightness in his right hamstring that forced him out of Sunday's victory. Snitker didn't have an update before the game on how much time he could miss. . Snitker said LHP Cole Hamels, the team's big offseason pickup who hasn't pitched yet this season due to triceps tendinitis, has been "throwing bullpens" and the next step was "live BP."
Red Sox: Placed LHP Darwinzon Hernandez on the 10-day injured list with a sprained left AC joint. . DH J.D. Martinez was out of the lineup after getting hit on the left hand by a fastball in the previous game. X-rays taken after Sunday's game were negative.
IT'S WHAT WE GOT
Snitker was asked if he was "disappointed" before the game that the club failed to make any moves at the trading deadline.
"No. We've been a pretty good team here for a month and a half," he said. "We'll see what we can do to try and win games."
Prospect Bobby Dalbec struck out in all four at-bats after homering in his major league debut Sunday.
After trading away OF Kevin Pillar (to Colorado) and LHP Josh Osich (Cubs) and placing Hernandez on the IL, the Red Sox brought up three players.
The 27-year-old Leyer made his major league debut after 236 minor league appearances over four organizations. He pitched an inning, giving up a run, but struck out Marcell Ozuna for his first career K.
Braves: RHP Ian Anderson (1-0, 1.50 ERA) is set to start the middle game of the three-game series Tuesday.
Red Sox: RHP Ryan Weber (0-2, 6.00) is slated to go for Boston.
More AP MLB: https://apnews.com/MLB and https://twitter.com/AP-Sports
Sticky stuff 101 - Everything you need to know as MLB's foreign-substance crackdown begins
Beginning Monday, Major League Baseball is ramping up efforts to stop pitchers from doctoring baseballs. Here's how the enforcement will work.
A leap of faith: How the Padres believed spending big would make San Diego a sports town again
With their ballpark at full capacity and a championship-caliber roster on the field, is it finally time to say goodbye to Dodger Stadium South?
Reds look to end 4-game losing streak against Twins
Cincinnati Reds (35-35, fourth in the NL Central) vs. Minnesota Twins (30-41, fourth in the AL Central)
Diamondbacks look to break 17-game losing streak against Brewers
Milwaukee Brewers (40-32, second in the NL Central) vs. Arizona Diamondbacks (20-53, fifth in the NL West)
Darvish expected to start for San Diego against Los Angeles
Los Angeles Dodgers (44-27, second in the NL West) vs. San Diego Padres (42-32, third in the NL West)
Rangers look to end 6-game losing streak against Athletics
Oakland Athletics (44-29, second in the AL West) vs. Texas Rangers (25-46, fifth in the AL West)