Philadelphia set a season high with 18 hits. Five Phillies had three hits.
"It's a good feeling because you put in the work," said Marlon Byrd, who drove in three runs with three hits, including a homer.
"It seems like you've put in the work every single day and don't get the results. The results are the key. You start feeling good about yourself.
"To come out here the whole series and put up runs against a good staff is huge for us, knowing that the work we've put in is starting to show up on the field."
Howard, who hit homers in each of the first two games of the series, drove in a run with a double in the Phillies' five-run second inning. He added a run-scoring single in the fifth. He had five hits, four walks and six RBIs in the series in Atlanta, where he has a good history.
Howard has 46 career homers against Atlanta, more than against any other team. His 20 homers at Turner Field are his high mark for any road stadium.
Asked what made the difference for Howard in the series, Phillies manager Ryne Sandberg said "Possibly just some confidence."
"This is a place he's hit well so maybe it brought back some positive thoughts," Sandberg said. "He has been seeing the ball real well, made them throw strikes and used the whole field."
Hernandez overcame his rough first inning to last six innings, giving up five runs and eight hits and five walks. He had been 0-4 in eight starts since his last win on May 4 against Washington.
"He had a rough first and the offense bounced right back for him and picked him up there," Sandberg said.
Harang (5-6) allowed nine runs, eight earned, in five innings. He threw 115 pitches while allowing 13 hits and three walks.
The Phillies have won seven of nine. Atlanta has lost eight of 12.
The Phillies' sweep knocked the Braves out of first place in the NL East, just in time for a visit to first-place Washington.
"We're going to go into D.C. with a little bit of a chip on our shoulders," Doumit said. "This wasn't a good series. It left a bad taste in our mouths, but we're going to go take it out on the Nationals."
Braves manager Fredi Gonzalez tried a new lineup, and rookie Tommy La Stella was 0 for 4 with a walk in his first game hitting leadoff.
"I thought we swung the bats well," Gonzalez said. "We were knocking on the door there almost the entire game trying to get back in it, which is a good sign."
Trailing 4-2, the Phillies scored five runs in the second on six straight two-out hits, including a two-run single by John Mayberry Jr., who had three RBIs.
Shortstop Cesar Hernandez, batting leadoff for the first time this season, had three hits and scored two runs. Domonic Brown and Will Nieves also had three hits. Nieves left the game with what Sandberg said appeared to be a quadriceps injury.
Phillies CF Ben Revere (sore left knee) had a pinch-hit groundout in the eighth. ... The Braves rested several starters including LF Justin Upton, who left Tuesday night's game with dizziness. CF B.J. Upton and SS Andrelton Simmons also received days off. ... Braves LHP Jonny Venters was given a platelet-rich plasma injection in his surgically repaired left elbow. He is trying to come back from his second Tommy John surgery on May 16, 2013, and will rest the elbow for about five weeks. ... Braves RHP Gavin Floyd faces Washington RHP Jordan Zimmermann to open a four-game series on Thursday. ... The Phillies open a four-game series at St. Louis with David Buchanan opposing Shelby Miller in a matchup of right-handers.