"We know that all the things that are happening right now for us are going to turn around," Prado said. "This kind of game is what we need to start something good."
Atlanta had gone 253 games without a grand slam before Prado's in the sixth. The Braves had also gone four days without a homer before Troy Glaus led off the inning with a shot off Brewers starter Doug Davis (1-4).
"We felt relieved after that," Prado said of Glaus' shot. "Tommy Hanson was great, but we knew we needed a couple of runs to support him."
So they gave him six in the inning.
After Glaus' homer and a fielding error by second baseman Rickie Weeks that let another run score, Davis walked the bases loaded with two outs. Prado, who had a career-high five RBIs, then hit reliever Manny Parra's fastball into the left-field bleachers.
"It's part of the game, everybody makes errors," Prado said. "After that, we had a pretty idea of where we were going."
Hanson (3-2) said he "didn't feel tired one bit" and would've been willing to try to complete the game after scattering four hits and being staked a big lead.
"It's always easier to pitch when you have some run support," the promising right-hander said. "It's definitely a good win for us."
Atlanta led 7-0 despite just five hits through the sixth and finished with seven hits, but it qualified as an offensive outburst for the Braves, who featured seven hitters in their lineup batting .252 or lower entering the night and three regulars at .192 or less.
Glaus' homer to start the sixth was Atlanta's first in a span of 169 at-bats. It would get even better with Prado's first career slam and the Braves' first since July 25, 2008.
"It was a pretty good pitcher's duel going there until the sixth inning," Brewers manager Ken Macha said. "That's when it kind of unraveled."
But Hanson froze McGehee on an 84 mph breaking ball that buckled the third baseman's knees and left fielder Matt Diaz made a nice leaping grab of Jim Edmonds' drive to the warning track that saved at least a run. Braun left before the seventh with a bruised left elbow and is day-to-day.
Davis was wild, walking six batters and hitting another. Prado's RBI single in the fifth was the first hit Davis allowed after a leadoff walk in the inning came back to haunt the left-hander.
"I'm puzzled," Davis said. "But I believe everything evens out, too."
Atlanta is still missing regulars Jason Heyward (groin) and shortstop Yunel Escobar (groin). Chipper Jones (groin) returned to the lineup, going 1 for 3 with a walk before leaving for a pinch runner in the sixth.
Jones had hustled down the line for an infield single before exiting. Cox said he wasn't sure if Jones would be available Tuesday after he was limping from the effort.
"I don't think it's any worse, to be honest," Jones said.
Heyward is expected to return Tuesday. ... Braves RHP Jair Jurrjens (hamstring) said he heard a pop while sprinting and will have an MRI back in Atlanta when the Braves return after this series. ... Brewers CF Carlos Gomez (strained left rotator cuff) was put on the 15-day DL after the game, retroactive to May 6. OF Adam Stern was called up from Triple-A Nashville. ... Brewers RHP David Riske will begin a rehab assignment at Single-A Brevard County on Tuesday.