Stanton hit his 37th homer, and Brantly hit a tiebreaking RBI double in the eighth inning to help the Marlins beat New York 3-2 Monday night.
"Very exciting, very well played. We pitched well, we got a couple hits when we needed. We got big hits today. We don't have to pitch a shutout to win games," Marlins manager Ozzie Guillen said. "It's nice to see the guys go out there and perform and don't rely on Stanton to do the damage."
Heath Bell (4-5) pitched a scoreless eighth inning and Steve Cishek pitched the ninth for his 15th save in 19 chances. Cishek issued a two-out walk to Fred Lewis before Ike Davis grounded out to end the game, giving the Marlins their second win in 11 games.
"In a situation with a man on second and nobody out you're hoping you can get a pitch to hit to the right side but with two strikes I'm going to try and battle my AB out and survive and get a good pitch up in the zone," Brantly said. "It was away, but I got a good piece of bat on it."
The Marlins trailed 2-0 before Stanton's solo home run off Manny Acosta in the sixth. It was Stanton's third home run in four games after missing the previous nine games with an oblique strain.
Jose Reyes hit a sacrifice fly in the seventh to tie the game at 2.
Mets right-hander Jeurys Familia made his first career start and did not allow a run in four innings. He walked six and struck out three.
"That's been his problem, base on balls," Collins said. "You saw when he was over the plate they don't hit them."
Marlins starter Wade LeBlanc allowed one unearned run in five innings.
Marlins RHP Josh Johnson was expected to start, but was replaced by LeBlanc due to a sore hamstring. ... Mets manager Terry Collins announced that he intends to bring back the entire coaching staff for next season. ... On Tuesday, Mets RHP R.A. Dickey (20-6, 2.69 ERA) will look to improve to 6-0 against Miami this season. The Marlins will counter with RHP Jacob Turner (1-4, 3.86 ERA).