The Marlins gave up three runs in the first, then rallied to win for the eighth time in 10 games.
Miami scored single runs in the second, third and fourth innings to tie the contest, Marcell Ozuna accounting for two of those scores with a solo homer and an RBI single.
The game remained deadlocked until the ninth, when the Marlins scratched out a run thanks chiefly to the wildness displayed by Atlanta closer Craig Kimbrel.
Both Ed Lucas and Morgan Morrison drew a walk off Kimbrel (2-2) and with two out Solano's base hit to right brought home Lucas from second base.
The Braves then put their leadoff man on first in the bottom of the ninth, but Steve Cishek retired the next three batters for his 17th save.
A.J. Ramos (3-2) collected the win, giving up one hit and striking out four while working the seventh and eighth innings.