Delgado (1-1), making just his sixth career start, earned his first major league victory, handling the Marlins on five hits and two walks while striking out one.
The 21-year-old right-hander's strong start enabled the Braves to reduce their magic number for winning the National League wild card to six with seven games remaining.
Cristhian Martinez and Anthony Varvaro each tossed two shutout innings of relief to complete the victory.
Anibal Sanchez (8-9) was saddled with the loss, yielding three runs on five hits over five innings for the Marlins, who couldn't maintain their momentum from Monday's come-from-behind 6-5 win over the Braves.
Swim Week Miami Beach 2014 [PHOTOS]