The first win by the Braves franchise came back on April 22, 1876, a 6-5 win over the Philadelphia Athletics (not the current Athletics franchise; this one folded after the 1876 season). This accomplishment was over 135 years in the making.
To commemorate the Braves' milestone, I created the timeline below. Click on the image to enlarge it.
The Giants (10,490 wins) and the Cubs (10,278 wins) are the only other teams to win 10,000 in the National League. The Reds (9,961 wins) should get there early next year. The Pirates (9,858 wins) should also reach 10,000 in the next few seasons.
The Dodgers and Cardinals have also won more than 10,000 games if you include their time in the American Association (Baseball-Reference counts the AA as a major league, but many other sources do not).
The Yankees have 9,723 wins, which is by far the most of any American League team. The AL didn't begin until 1901, so the NL teams got as much as a 25-year head start on the Yanks. The Yankees actually have won over 500 more games than the Phillies, despite the fact that the Phillies started 18 years earlier.
Speaking of the Phillies, they are the only franchise to lose 10,000 games (they're at 10,266). They won't be alone in that unfortunate club for long, though: the Braves are only 8 losses away from 10,000.