Johnson was 2-for-4 with a two-run double and a run-scoring single. Atlanta won for only the third time in 13 games not started by Jones, whose .394 batting average leads the majors.
"I was just happy, when you're hitting in Chipper's spot, that you get a chance to drive in runs," Johnson said. "When you hit in the middle of the order, that's what you want to do."
Jones, recovering from a strained right quadriceps, struck out in the eighth as a pinch-hitter. He said he feels better and hopes to be the designated hitter at Toronto beginning on Friday night.
"I'm only feeling it when I stretch," Jones said.
Atlanta has other injury concerns.
Shortstop Yunel Escobar missed the game after jamming his left shoulder during Tuesday night's 4-3 loss. His replacement, Omar Infante, left in the second inning after straining his right hamstring while running out a double. Rookie Brent Lillibridge came in for Infante.
Ruben Gotay, who started at third base for Jones, was 0-for-4.
Outfielders Mark Kotsay and Matt Diaz and backup infielder Martin Prado are on the disabled list. Starter John Smoltz had season-ending shoulder surgery this month, and the Braves don't know when starter Tom Glavine and closer Rafael Soriano will return from elbow injuries.
Still, the Braves entered Wednesday night only four games behind first-place Philadelphia in the NL East.
"I think everybody in here feels at this point in the season we have to pick it up," Johnson said.
"We're not that far out of it. Let's win these games and get everybody healthy and then see what happens," he said.
Campillo (3-2) gave up two runs and four hits in seven innings and walked none. He has a 2.54 ERA but had been 0-2 in his previous five starts since winning at Milwaukee on May 29.
"He's been excellent every time out," manager Bobby Cox said. "He's legit. When we stuck him in the rotation we didn't know for sure, but he sure opened our eyes in relief. ... He looks like a classy veteran pitcher to me."
Milwaukee has scored a combined three runs in two losses to Campillo.
"He is so consistent," Ryan Braun said. "He doesn't make many, if any, mistakes. He's been pitching like that all year. He has great numbers. He's a competitor."
Blaine Boyer pitched around Rickie Weeks' two-out triple in the eighth, and Mike Gonzalez finished for his second save in two chances since coming off the disabled list on June 18. Braun singled leading off against Gonzalez, but Prince Fielder grounded into a forceout and Corey Hart and Bill Hall struck out.
Jeff Suppan (4-6) gave up four runs and nine hits in six innings. He is 0-2 in three starts since June 8.
Milwaukee had won its previous four road games. The Brewers are 14-23 at Turner Field and have never swept a series in Atlanta.
Run-scoring doubles by Mike Rivera and Weeks put Milwaukee ahead in the third. Atlanta, which scored two runs in the first two games of the series, tied it in the bottom half on Johnson's two-run double. The Braves had been 0-for-10 with runners in scoring position in the series before Johnson's hit.
Gregor Blanco's RBI single put Atlanta ahead in the fifth, and Johnson followed with an RBI single. Blanco drove in Brandon Jones, whose line-drive double sailed over Braun in left field after Braun took a first step in.
"That is hard to read," Braun said. "It's tough to see the ball back there."
Brewers right-hander Eric Gagne pitched a scoreless inning in his first rehab assignment with Triple-A Nashville on Tuesday night. Gagne, on the DL with right rotator cuff tendinitis, is scheduled to pitch an inning Thursday night. ... Boyer made his 42nd appearance, the most in the majors. ... Infante will be examined Thursday but is not expected to be placed on the DL. ... Lillibridge hit an eighth-inning double off Mitch Stetter for his first career hit.