And the Braves did have a pretty good chance, until Los Angeles' newest sensation, Yasiel Puig, blew the game open in the eighth.
Atlanta's bullpen had allowed just one run in its last 27 2/3 innings, but Cory Gearrin struggled to get outs. Gearrin loaded the bases via singles to Skip Schumaker and Luis Cruz and a walk to Hanley Ramirez, who was pinch-hitting for the pitcher's spot.
Puig then came up and wasted no time as he hit Gearrin's first pitch -- an elevated slider -- over the right-field wall for his first grand slam, his third home run in his brief four game career.
"I feel like he feels he belongs here and he's just proving it right now," Dodgers starter Zack Greinke said. "I don't think in his mind he's worried about the pitcher being better than him. He feels he belongs."
While the script of Puig's first week in the Majors has been fitting for Hollywood, he is quickly showing everyone that he does in fact belong.
"Didn't make my pitch, left it up and out over the plate and he did what he was supposed to do," Gearrin said. "You tip your cap to him, but it's something I'm definitely going to learn from. Have to execute better in that situation."
While the Braves got a tremendous start from Hudson, the Dodgers received an even better one from Greinke, who shut out Atlanta for seven innings.
In his longest start of the season, Greinke allowed four hits and struck out seven, a season high.
"He's Zack Greinke, he got paid for a reason," Braves right fielder Jason Heyward said. "He has five above-average pitches. He's got his fastball -- four-seam and two-seam -- he's got his cutter, he's got a good changeup, he's got a curveball and he makes his pitches when he needs to."
The Dodgers' first run came when left fielder Scott Van Slyke led off the second with a single and Andre Ethier followed with a double that dropped between Heyward and center fielder B.J. Upton.
With runners on second and third and no outs, the Dodgers appeared primed for a big inning, but Hudson limited the damage to just one run -- an RBI groundout from Schumaker.
However, that one run was all Greinke needed.
Los Angeles' right-hander allowed baserunners in five of his seven innings, but only one reached third base.
Atlanta's biggest scoring threat came in the seventh, when Freddie Freeman and Brian McCann led off the inning with hits, giving the Braves runners on the corners with no outs.
But, just as he did throughout the evening, Greinke neutralized the Braves' bats. Greinke struck out Dan Uggla then got Ramiro Pena and Upton to fly out.
"Those situations come back and haunt you every single time," Gonzalez said.
Despite the strong outing, Hudson has lost four straight decisions. However, Thursday was his second consecutive quality start and the veteran pitcher feels he has turned a corner.
"I felt pretty good," Hudson said. "It's just staying tall over the rubber, not overthrowing and staying behind all my pitches and trusting it. That's what it boils down to."
His manager agrees.
"That's a couple outings in a row where he's given us a real good pitching performance," Gonzalez said. "I can't wait for him to go out there in another five days and give us another chance to win a ballgame."