Swanson had been 0-for-8 since returning from the disabled list on Saturday. He drove a 2-2 pitch from reliever Tayron Guerrero into the left field corner to bring home Tyler Flowers and Kurt Suzuki with the tying and winning runs.
Swanson struck out to start the inning, but the Braves began to peck away against reliever Brad Ziegler. Ronald Acuna Jr. drove in a run with a sacrifice fly,
Freddie Freeman's hustle on an infield single plated another, and Nick Markakis drove in a run with a single to center field.
Miami brought in Guerrero, who walked Flowers and surrendered an RBI hit to Suzuki. He walked Johan Camargo to load the bases with two outs, setting up Swanson's game-winning heroics.
Guerrero (0-2) took the loss, giving up two runs on two hits and two walks, failing to retire a batter.
A.J. Minter (3-0) recorded the win, despite allowing a run in his one inning of work.
The Braves won two of three games in the series and improved to 7-0 in rubber games. Atlanta is 5-2 against Miami.
The rally deprived Miami's Wie-Yin Chen of his first victory over Atlanta since 2012. The southpaw allowed two runs in 5 1/3 innings, giving up five hits, one walk and striking out five.
Atlanta starter Julio Teheran gave up six runs in five innings on seven hits and three walks. He struck out four. Teheran has allowed 10 runs in his last 11 innings.
Miami's Lewis Brinson hit his sixth homer, his first career grand slam, and Miguel Rojas was 3-for-3 with four RBIs. Rojas hit two home runs, his sixth and seventh of the season.
Freeman was 3-for-5, while Ozzie Albies, Acuna and Markakis each had two hits.
Atlanta made a roster move on Sunday, granting infielder Jose Bautista his release and recalling right-handed pitcher Lucas Sims from Triple-A Gwinnett.
Bautista, signed last month to a minor-league free-agent deal, hit only .145 in 12 games. The Braves opted to go with Camargo as their regular third baseman and cut ties with Bautista.
--Field Level Media
Miami | Atlanta | |
Wei-Yin Chen | Player | Julio Teheran |
No Decision | W/L | No Decision |
5.1 | IP | 5.0 |
5 | Strikeouts | 4 |
5 | Hits | 7 |
3.38 | ERA | 10.80 |
Miami | Atlanta | |
Miguel Rojas | Player | Freddie Freeman |
3 | Hits | 3 |
4 | RBI | 1 |
2 | HR | 0 |
9 | TB | 3 |
1.000 | Avg | .600 |