Jones' big hit saved the night for the Marlins, as they nearly let an opportunity to gain ground on the Braves slip away.
The Marlins have won two straight for the first time since July 5-6, and they moved to 7 1/2 games behind Atlanta in the National League East by claiming the first of four games at Turner Field.
"It definitely felt good to get one and help the team win late like that," Jones said. "It's been a grind. We've been grinding out games like that. There was great pitching on both sides. It was going to take one hit to pretty much win the game. I'm glad I was able to come through finally, for the team. It's been a while. I had been scuffling, personally."
In the 10th, Christian Yelich singled off of Shae Simmons, and moved to second on Jordany Valdespin's sacrifice bunt. Giancarlo Stanton was intentionally walked. Casey McGehee drew a walk, loading the bases for the second straight inning for Miami. This time, Jones came through.
The Marlins filled the bases with no outs against Craig Kimbrel in the ninth inning, only to be turned away.
With no margin for error, Kimbrel got out of it, striking out Jarrod Saltalamacchia and Adeiny Hechavarria, who fanned four times on the night. The threat ended when pinch-hitter Donovan Solano, on Kimbrel's 27th pitch of the inning, lined out to second.
"It's tough when you're facing the best closer in baseball," Jones said. "You've got to tip your cap to him. He made pitches. He's tough. Nothing is easy when it comes to facing him."
Steve Cishek nailed down his 22nd save in 25 opportunities, striking out two in a perfect 10th. Miami is now 6-9 in extra innings.
"Every game for us is big from here on out," Miami manager Mike Redmond said. "We're behind. We've got a lot of ground to make up. I also believe we're a better team when we're playing our game, relaxed and having fun. And not putting pressure on ourselves."
Atlanta had its chance in the ninth against Bryan Morris, but came away empty. Andrelton Simmons doubled with two outs, and pinch-hitter Ryan Doumit walked. But the inning came to a close on B.J. Upton's lineout to right.
Tom Koehler threw 6 2/3 innings, allowing one run while striking out seven. Miami needed a strong start because Atlanta All-Star Julio Teheran was dominant, matching his career high with 11 strikeouts.
"Any time you go out there and face Julio, you know runs are going to be at a minimum," Koehler said. "So you want to go out there and match as many zeros as you can, and try to get the team in the dugout as quickly as possible."
Teheran held Miami hitless for 4 2/3 innings before Saltalamacchia's no-doubt home run, his 10th of the season.
In the fourth inning, the Marlins were denied a hit on a spectacular play by Simmons at short to rob Valdespin of a sure single to center.
Yelich became Miami's first baserunner when he walked to open the fourth. On a hit-and-run, Valdespin stung a sharp grounder up the middle. Simmons was moving to cover second, and lunged backwards to snare the grounder. From the ground, he spun and tagged second base to force out Yelich.
The Braves grabbed the early lead, pushing across a run in the second inning on Chris Johnson's RBI groundout to first.
"He was good," Johnson said of Koehler. "But it also stinks as an offense because Teheran did so good. So, we kind of kick ourselves and wish we could have gotten enough runs."
Koehler avoided more damage in the second when he struck out Evan Gattis for the first out, and Johnson followed with his run-scoring groundout. Simmons tapped to third to close out the inning.
"It's a huge series," Koehler said. "I don't want to say it's make or break, but we're running out of season. These are guys who are ahead of us."