Greinke scattered three hits, striking out three and walking four, and received support from Hanley Ramirez's two-run homer in the fourth.
For the Phillies it was not the start they would have wanted for new manager Sandberg, who had replaced long-time skipper Charlie Manuel earlier in the day.
While that was the Dodgers' 24th win in 27 games since the All Star break, the Atlanta Braves stretched their run to 18 in 20 as Justin Upton's walk-off homer in the 10th helped them see off the Washington Nationals 3-2.
Jayson Werth's RBI single in the eight had sent the game into extras, after a pitching duel involving two impressive rookies, the Nationals' Taylor Jordan and Alex Wood of Atlanta.
Jordan threw six scoreless innings, while Wood struck out nine in 6 1/3 frames of one-run ball.
Alfonso Soriano's torrid form continued as he drove in four runs on three hits, including a homer, as the New York Yankees beat the Boston Red Sox 10-3.
In his last four games, Soriano, acquired last month from the Chicago Cubs, is 13-for-18 with five homers, nine runs and 18 RBIs.
Another recently-acquired player, Mark Reynolds, also shone, driving in three runs on a pair of hits, including a long ball.
Eric Hosmer launched a two-run homer and James Shields pitched seven scoreless innings to help the Kansas City Royals complete a sweep of their day-night double-header against the Detroit Tigers with a 3-0 win.
Danny Duffy's six shut-out frames had led them to a 2-1 win in game one.
Jose Lobaton's ninth-inning RBI triple lifted the Tampa Bay Rays to a 5-4 walk-off win over the Toronto Blue Jays.
Hunter Pence had five RBIs while Brandon Belt and Pablo Sandoval had four hits apiece as the San Francisco Giants came out on top in a wild game against the Miami Marlins, sealing a 14-10 win.
Jake Arrieta threw seven scoreless innings for his first win with the Chicago Cubs as they beat the St Louis Cardinals 7-0.
Wilin Rosario launched a two-run go-ahead homer in the seventh to lead the Colorado Rockies to a 6-3 victory over the Baltimore Orioles.
Jonathan Lucroy hit a two-run walk-off homer as the Milwaukee Brewers beat Cincinnati 7-6 to snap the Reds' winning streak at five.
Yoenis Cespedes cracked a two-run shot in the first to set the Oakland Athletics on their way to a 3-2 win over the Cleveland Indians.
Adam Dunn and Jeff Keppinger went yard in the Chicago White Sox's 5-2 win over the Minnesota Twins, while Kyle Seager's two-run shot lifted the Seattle Mariners to a 3-1 victory over the Texas Rangers.
The Pittsburgh Pirates beat the Arizona Diamondbacks 6-2, the New York Mets saw off the San Diego Padres 5-2 and home runs in the ninth inning from Chris Carter and Matt Dominguez helped the Houston Astros to an 8-2 victory against the Los Angeles Angels.