The 10th began with placement runner Jon Berti advancing from second to third when Braves second baseman Ozzie Albies ran backward to try to catch Starling Marte's shallow fly, but had the ball bounce off his glove for a single. Berti gave the Marlins their first lead when Cooper's single landed in center field to make it 4-3.
Brian Anderson followed with an RBI double, and the Braves couldn't muster a rally in the bottom of the inning.
Yimi Garcia (1-1) pitched the final 1 1/3 innings. Webb (0-1) took the loss.
Ronald Acuna Jr. tripled and scored two runs, and Huascar Ynoa pitched a career-high six strong innings for the Braves, who have lost two straight.
The teams were facing each other for the first time since Atlanta swept Miami in their NL Division Series last year.
''They're the upper echelon of this division,'' Cooper said. ''It's good to come in here and take the first one and hopefully it continues and keeps everyone's morale high to continue into the next few days.''
Acuna flashed his speed in the third, hitting a triple into right and scoring on a shallow sacrifice fly by Albies. Acuna slid across the plate headfirst after second baseman Jazz Chisholm Jr. caught the ball over his shoulder and was backpedaling when he stopped to throw toward the plate.
Acuna pumped his fist toward the Atlanta dugout in celebration after scoring to make it 2-1.
Ynoa struck out a career-high 10 while allowing one run and three hits. The rookie struck out every batter he faced at least once except Miguel Rojas.
Atlanta led 1-0 in the first when Acuna walked, stole second and scored on Marcell Ozuna's single.
Adam Duvall hit his second homer to begin the second, connecting against his former team to tie it 1-all.
Miami starter Sandy Alcantara gave up three runs, four hits and three walks while striking out four in 6 1/3 innings. He was chased after walking pinch-hitter Pablo Sandoval and replaced by Anthony Bass, who walked Acuna to load the bases and gave up Albies' sacrifice fly to make it 3-1.
Atlanta reliever A.J. Minter walked three in the eighth and left his replacement, Nate Jones, with runners on first and second and one out. Cooper, who had been hitless in his last 13 at-bats before singling in the sixth, followed with a two-run double to make it 3-all.
''You definitely don't want to be chasing and get yourself in too big of a hole,'' Marlins manager Don Mattingly said of the NL East race. ''I do think the Braves are the measuring stick in this division. Obviously there's a lot of good teams, but this is one of those humps that we have to get over. That doesn't mean you're going to go out and beat these guys 15 times, but we've got to hold our own a little better than we have in the past.''
The Braves' offense is still collectively struggling despite Acuna's recent hot streak. Atlanta went 1 for 11 with runners in scoring position, and none of its Nos. 2-6 hitters Monday were batting above .189.
''I think we're capable of close to what we did last year,'' manager Brian Snitker said. ''Once we get this thing going - but sometimes it doesn't work that way. You just have to keep grinding through it. Eventually, if you handle tough situations, there's good on the other end of them. Always.''
Miami C Jorge Alfaro was scratched from the lineup because of lingering hamstring tightness. Mattingly said he hopes Alfaro, who has missed five straight games, will be back by Thursday. ... Marlins RHP Sixto Sanchez (shoulder discomfort) is progressing at the team's alternate training site in Jupiter, Florida. He is undergoing rehab exercises and throwing at 45 feet. ... RHP Jorge Guzman (elbow) is throwing bullpens in Jupiter and should face live hitters next. ... RHP Edward Cabrera (biceps) is throwing at 75 feet. ... RHP Elieser Hernandez (biceps) has yet to start throwing.
Mattingly was thrown out after a call was overturned in the eighth that originally had Berti ruled safe at second on a stolen base attempt. It appeared Berti beat shortstop Dansby Swanson's tag attempt.
Braves LHP Max Fried (0-0, 9.00 ERA) faces RHP Pablo Lopez (0-1, 1.54) in the second of a four-game series. Fried is 0-1 with a 5.13 ERA in six career starts against Miami. Lopez is 2-4 with a 4.43 ERA in eight career starts versus Atlanta.
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