Skip Schumaker had four of the Dodgers' 12 hits, and the club improved to 19-3 since the All-Star break. Los Angeles is a season-high 16 games over .500 after winning four straight.
"It feels pretty good right now, the way we're playing," said manager Don Mattingly.
The Dodgers are 36-8 since June 22, the best record in the Major Leagues during that span. It's also the best 44-game stretch for the franchise since 1953.
"Things have been going well for us now for a couple months," Punto said. "Every day you wake up and are excited to get to the yard. It's just a great atmosphere."
Greinke won his 10th game of the season, and is 6-1 with a 2.32 ERA in 10 starts at home. He struck out seven against one walk in his fifth scoreless outing of the year.
"Everyone is playing good, I'm just feeding off the energy," said Greinke, who has six wins over the club's historic 44-game stretch. "For the most part, I'm executing pitches and all of them are a lot sharper."
The Dodgers staked Greinke to an early lead, courtesy of two players who figured prominently in Friday night's comeback from a six-run deficit.
Punto legged out an infield single in the first inning with a head-first slide into the bag. Gonzalez followed with a two-run shot to right field. The first baseman leads the Dodgers with 16 homers and 72 RBIs.
In Friday's four-run ninth inning, Gonzalez drove in Punto to tie the game and later scored the winning run.
Punto singled again in the third to score Greinke and Carl Crawford. Greinke drew a leadoff walk before Crawford laced a double down the right-field line, his third in his past four starts.
From there, Greinke was in control until the fourth, when he pitched out of trouble after a leadoff double and a walk put two on with one out. Greinke induced a ground ball, but the Dodgers couldn't turn a double play. The right-hander didn't unravel, and promptly got a comebacker from Kelly Johnson to end the frame.
"Zack is one of the best pitchers in the league, and he showed why today," Gonzalez said. "We gave him a lead early and let him do his thing."
Juan Uribe and the Dodgers were the victims of a hidden-ball trick in the fourth after Uribe tagged to third on a sacrifice fly. Former Dodgers first baseman James Loney started the play from just behind the mound, throwing to shortstop Yunel Escobar, who fed third baseman Evan Longoria. Uribe drifted off the bag at third, and Longoria applied a tag for the second out of the inning.
Neither Uribe nor third-base coach Tim Wallach knew Longoria, standing several feet behind the bag, had the ball.
"I was watching it, and I didn't know what to do to stop it," said Greinke, who saw it all unfold from the on-deck circle.
Greinke has made seven consecutive quality starts and is 4-1 with a 1.66 ERA since July 8. The Dodgers are 9-1 over his past 10 outings.
"He's kind of one of those guys who hits his spots early in the game and doesn't really mix his pitches a whole lot until he needs to," Longoria said. "And then he goes to the well and all of his pitches are plus."
J.P. Howell came on for Greinke with two on in the seventh and got a strikeout and a groundout to help preserve the shutout. Chris Withrow followed with two perfect innings to secure the victory.
The Dodgers have 14 shutout victories this season, tied with the Pirates for the most in the National League.
"We've got some good starters over here and the bullpen does it, too," Greinke said.
Punto has filled in well at shortstop for Hanley Ramirez over the past week while Ramirez continues to nurse a sore right shoulder. Punto is hitting .500 (9-for-18) with three doubles and five RBIs over his past seven games.
Schumaker also has excelled in a reserve role and has three straight multihit games. He's hitting .643 (9-for-14) with five RBIs over his past four games.
"The good thing about Skip and I, we love to play baseball," Punto said. "We love to get here and see our name in the lineup. That's not really our role to come here and be starters and play every day. But when we do see our name in the lineup, it's why we still play and love the game."