The left-hander keeps winning, too.
"Today wasn't one of his best outings," manager Fredi Gonzalez said. "He was spraying it around a little bit and got his pitch count up, but he's tough to take swings at."
The Braves have won two straight, improving the majors' best record to 79-52. They have won 14 of 17 at Turner Field and own baseball's home mark at 45-18.
Closer Craig Kimbrel pitched a perfect ninth to earn his 42nd save, most in the majors, in 45 chances.
Atlanta took a 2-0 lead in the second inning. Brian McCann walked with one out, advanced to third on Joey Terdoslavich's single to right field, and both runners scored when Johnson tripled over Stubbs near the wall in right.
Wood (3-2) allowed five hits, a career-high four walks and struck out five in 5 2/3 innings. He's 3-0 with a 1.46 ERA in his past six starts.
Wood credited pitching coach Roger McDowell with making a timely mound visit in the third. The rookie retired the first two batters he faced and was ahead in the count on Jason Kipnis and Carlos Santana before walking both batters.
McDowell wanted to help Wood slow down his pace and not rush to make the next pitch. But he also wanted to talk strategy with the rookie and McCann, the catcher.
After McDowell returned to the dugout, Wood got Mike Aviles to ground into a forceout and end the threat.
"He comes out and he has an idea what we should do to that hitter," Wood said. "That's definitely a big help. As long as we execute it, it usually ends up working out pretty good."
Cleveland had won two straight and five of six, but lost despite holding Atlanta to just three hits.
"Our pitching's been on point, but you've got to score to win," Bourn said. "You can't score zero runs and win the game. It's not going to happen.
"We've got to find a way to put some runs on the board and help our pitchers out."
Rookie Danny Salazar (1-2) gave up three hits, two runs and two walks in four innings. The right-hander struck out three.
Wood issued his four walks in the first three innings, but he worked around Asdrubal Cabrera's leadoff double in the fourth and Kipnis' two-out double in the fifth.
A two-out single by Michael Brantley chased Wood in the sixth. The next Atlanta pitcher, Luis Ayala, walked Stubbs, the only batter he faced, but Scott Downs ended the threat by striking out pinch-hitter Jason Giambi.
Cabrera went 3 for 4 and was the only batter with more than one hit, but Johnson drove in the game's only runs.
"I got a fastball out over and hit it well enough to get it over Stubbs' head, and we get a couple of runs off a guy that throws really hard," Johnson said. "But Woody was the story of the day, pretty much commanding all of his pitches and keeping them off balance."
Gonzalez indicated that Johnson, claimed off waivers last week from Kansas City, likely will stay with Atlanta with second baseman Dan Uggla set to come off the disabled list on Wednesday.
Johnson started his fifth game at second base in Uggla's place.
Braves RF Jason Heyward rejoined the team for the first time since getting hit by Mets pitcher Jonathon Niese last Wednesday, breaking the right side of his jaw with a 90 mph fastball. Heyward hopes to return for the playoffs. ... Atlanta RHP Brandon Beachy was in the clubhouse one day after learning there's no structural damage in his surgically repaired elbow. Beachy, who has missed 109 games over the past two years, hopes to return by the end of next month. ... The Indians and Braves met for the first time since 2007 and for the first time at Turner Field since 2004. ... Uggla went 2 for 4 with a homer in his last rehab appearance Tuesday with Triple-A Gwinnett. ... Atlanta had its fewest hits in a victory since getting three on July 19, 2012, against San Francisco.