Ramon Hernandez started the rally with a one-out single, took second on the first wild pitch and advanced to third on a single by Luis Cruz.
Varvaro (3-1) then bounced another wild pitch to Juan Uribe, allowing Schumaker, who was running for Hernandez, to score. Puig was on deck, but he wasn't needed after his fourth homer in five career games tied it up in the sixth.
"I was just trying to get some balls below the zone to get a ground ball there, and it just skipped the wrong way," Varvaro said. "I'd rather it leave the yard, I guess. That's just how it happens sometimes."
The Dodgers have won four of five since Puig, a 22-year-old Cuban defector, was called up Monday from the minors. It was their second straight win over the NL East-leading Braves, who lost for the first time in six games Thursday.
With a runner on second in the 10th, Justin Upton hit a dribbler in front of the plate that catcher Hernandez fielded and fired to first base. Umpire CB Bucknor called Upton out to end the inning, although the replay clearly showed he had beaten the throw. Braves manager Fredi Gonzalez came out to argue to no avail.
"I didn't think it was as close as he thought it was," Gonzalez said. "We would have liked to get an opportunity with (Freddie) Freeman at the plate -- first and third and your hottest hitter up. But it didn't happen."
Freeman was 3-for-4 and has hit safely in 12 of his past 15 games.
The Dodgers had the potential go-ahead run on third with two outs in the eighth when reliever Jordan Walden intentionally walked Puig, the first time this week a team didn't want to pitch to the rookie. Mark Ellis flied out to end the inning.
Puig tied the game 1-all with his fourth homer and 10th RBI of the week with two outs in the sixth. He sent an 0-1 pitch from Paul Maholm into the left-field pavilion.
Puig was hitless in his first two at-bats.
"On the home run, to be honest, I missed by about 3 feet. I wasn't even trying to throw anywhere close to the zone," Maholm said. "But I hung one, and he's hot. He's a free swinger, so you've just got to make your pitches."
Schumaker said, "There's a different energy from the Cuban. He's just been amazing for us, and it feels like something is going to happen every time he comes up."
Puig tied the record for most RBIs through five career games, shared by Danny Espinosa (2010) and Jack Merson (1951), according to information provided to the Dodgers by Elias Sports Bureau.
"I try to prepare myself with the coaches, and it's just happening," Puig said through a translator. "As long as the team is pulling together, I'm happy to be a part of it."
The Braves led 1-0 on Dan Uggla's RBI single with two outs in the fourth.
Hyun-Jin Ryu returned to the mound after missing his previous start for the Dodgers. He bruised his left foot May 28 against the Angels, when he tossed the first shutout of his career in a 3-0 victory. The left-hander allowed one run and six hits in 7 2/3 innings, struck out six, and walked one.
Maholm gave up one run and four hits in 7 1/3 innings, struck out three, and walked two.
Puig is the second player since 1900 to hit four homers in his first five games, joining Mike Jacobs of the New York Mets in 2005, according to Elias. ... Dodgers INF Hanley Ramirez was out of the starting lineup for the second straight game Friday because of tightness in his left hamstring, but he pinch hit in the ninth. "For me it's a concern because it's two days," manager Don Mattingly said. The same hamstring landed Ramirez on the DL earlier this season. ... Dodgers LHP Ted Lilly is in doubt to start Sunday's series finale. He was bothered last time out by a chronic neck issue, and Mattingly said that if Lilly can't make his start, a minor leaguer will be called up. ... Braves RHP Brandon Beachy will make his fourth injury rehab start Saturday for Triple-A Gwinnett.
Copyright by STATS LLC and The Associated Press
LOS ANGELES, CA - JUNE 07: Outfielder Yasiel Puig #66 of the Los Angeles Dodgers signs autographs during batting...
47,164 (84.2% full) - % is based on regular season capacity
67 degrees, clear
Home Plate - Dale Scott, First Base - CB Bucknor, Second Base - D.J. Reyburn, Third Base - Todd Tichenor
Yasiel Puig hit a solo HR, giving him 4 career HR: a solo shot, 2-run HR, 3-run HR and grand slam...
The last Dodgers player whose first four career home runs were a solo shot, 2-run, 3-run and grand slam was Frank Howard whose first four career homers spanned 1958 to 1960.