Prado also homered, tied a career high with four hits and drove in three runs.
Michael Bourn doubled to right field off Edward Mujica (8-4) with one out in the 11th. Prado then singled to shallow right field and Mike Stanton's throw went over catch Brett Hayes' head, allowing Bourn to score.
George Sherrill (3-1) pitched a scoreless ninth to earn the win.
Braves starter Brandon Beachy struck out 10 and did not allow a run in 6 2/3 innings.
Uggla hit an infield single in the fifth inning off Clay Hensley to tie Andre Ethier for the majors' longest hitting streak of the season. The streak is the third-longest in Braves franchise history. Tommy Holmes hit in 37 straight games in 1945.
Uggla finished 1-for-6 against his former club and is hitting .220 on the season. He was batting .173 when his streak began, the lowest average for any hitter with more than 10 at-bats at the start of a 30-game streak since at least 1918, according to STATS LLC.
The Marlins have lost six straight since climbing to .500 for the first time since June 12.
Florida had a golden opportunity to score in the eighth after Eric O'Flaherty walked the bases loaded with one out. O'Flaherty then struck out Logan Morrison swinging and caught Cameron looking at a called strike three to end the threat.
Prado's 11th home run in the fifth gave the Braves a two-run advantage, and he had an RBI single in the sixth to stretch the lead to 3-0, ending Hensley's outing.
Hensley allowed eight hits in 5 2/3 innings.
It was the fifth time Prado had four hits in a game. The last time was May 16, 2010, against Arizona. ... Braves shortstop Alex Gonzalez extended his hitting streak to 10 games with a single in the second. ... Marlins shortstop Hanley Ramirez (shoulder) missed his sixth straight game and a stint on the disabled list is a possibility according to McKeon. ... Marlins All-Star first baseman Gaby Sanchez is hitting .215 after the break and was held out of the starting lineup. "I think he's trying too hard," McKeon said. ... The Braves will send right-hander Tim Hudson to the mound Wednesday. He is 10-4 in 21 starts against the Marlins, who will counter with Anibal Sanchez as he looks for his first win since June 10.
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