When Prado saw a changeup to his liking, he hit it hard.
"I was just going to put the ball in play," he said. "I was thinking any pitch I can drive, hit it solid. I saw it, I hit it."
The Braves avoided getting swept in a three-game series for the first time this season and gave themselves a good send-off for an eight-game road trip that begins Monday against NL East-leading Philadelphia.
"It was big for us not to go into Philly having lost three straight," third baseman Chipper Jones said. "One of the biggest wins of the year, I'll tell you."
Jose Constanza set up the win with a one-out single off Hawksworth (2-5). Costanza moved to second on a wild pitch and scored easily when Prado's liner landed inside the left-field line.
Craig Kimbrel (4-2) retired the side in order in the top half to get the victory.
The Braves, who began the day with a 7 1/2-game lead in the NL wild-card standings, put just one runner in scoring position through the first six innings against Clayton Kershaw.
But third baseman Aaron Miles' one-out throwing error in the seventh proved costly for Los Angeles.
The Dodgers had a 3-0 lead when Alex Gonzalez singled and moved to third on Jack Wilson's single. Miles fielded Constanza's hard grounder cleanly, but his wide throw allowed Gonzalez to score and ruined a potential double play.
Wilson and Constanza moved up on a wild pitch and scored on Brooks Conrad's broken-bat single, tying it at 3.
Kershaw, who was attempting to tie Arizona's Ian Kennedy for the NL lead with his 18th victory, allowed three runs, two earned, and 10 hits in seven innings. He struck out 10 and walked none.
Kershaw threw 84 of 115 pitches for strikes in his first career start at Turner Field. The left-hander was 4-0 with a sparkling 0.58 ERA in his previous four outings.
"I definitely felt like I had good stuff, some of the better stuff I've had all year," Kershaw said. "I had command of all three pitches. In the seventh, I left some (pitches) up. Wilson was a hanger and with Gonzalez I got behind in the count."
Matt Kemp went deep for the Dodgers, hitting a three-run drive with two out in the third for his 32nd homer.
Randall Delgado, the third rookie starting pitcher used by Atlanta in the series, gave up three runs and four hits in five innings.
"We haven't been swept in a three-game series all year and it wasn't looking good with Kershaw on the mound," said Jones, who walked as a pinch hitter in the ninth. "But we caught a break. They made a mistake on defense and Brooks Conrad capitalized and we were able to scrape one across there."
Los Angeles 2B Justin Sellers stayed in the game after Delgado hit him on the right wrist in the third inning. ... Braves RHP Tommy Hanson, on the disabled list with shoulder tendinitis, will try some long toss Monday in Philadelphia. Atlanta has no timetable for his return. ... Braves rookie OF Antoan Richardson, whose contract was purchased from Double-A Mississippi, singled in his first career at-bat as a pinch hitter in the fifth. ... Atlanta RHP Peter Moylan was activated from the disabled list after missing 123 games with a strained back. ... Wilson was making his Atlanta debut after getting traded from Seattle on Wednesday. ... Conrad, who landed on the first base bag awkwardly after his pinch-hit single, was walking with an air cast on his right ankle after the game. He doesn't believe the injury is serious.