Gonzalez, who had a two-run homer and five RBIs in the series opener Friday, followed that up with a shot to the left-field seats off Wade Miley (2-2) with two outs in the third inning.
Greinke (3-0) gave up a run on eight hits in 5 1-3 innings, striking out eight with no walks.
Miley, who has two of Arizona's three wins this season, gave up five runs and eight hits in five innings.
The Dodgers, who have won four of five, got the leadoff batter on base five times in the first six innings, three via walks, and four of them scored.
Los Angeles scored twice without an out in the second inning. Gonzalez drew a leadoff walk, Scott Van Slyke doubled to put runners at second and third, and Uribe singled to bring both of them home.
Puig led off the third with a single, then with two outs Gonzalez homered to make it 4-0 and give the slugger seven RBIs in the first 12 innings of the series.
Los Angeles added a fifth run in the fourth. Justin Turner led off with a double and scored on Puig's two-out single.
Arizona finally got a run after being blanked for the first 13 innings of the series when Pollock led off the fifth with a home run to the porch in right-center, his first career pinch-hit homer.
The Dodgers got an unearned run without a hit in the sixth. Oliver Perez walked Drew Butera to start the inning. Greinke's bunt advanced the runner to second, then Butera scored when third baseman Martin Prado couldn't handle Matt Kemp's sharp grounder.
The Dodgers piled on another run in the ninth. Uribe doubled a couple of inches below the home run line on the left field wall, and eventually scored when pinch-hitter Andre Ethier ground into a fielder's choice.
NOTES: Gonzalez has 26 career home runs and 84 RBIs against the Diamondbacks, most of any active player. ... With the left-hander Miley on the mound, the Dodgers sat left-handed batters Ethier and Carl Crawford. ... The Dodgers have Dan Haren (2-0, 0.75 ERA) on the mound Sunday while struggling Trevor Cahill (0-3, 7.90) goes for Arizona. ... The Diamondbacks have lost four of five series to start the season. ... Los Angeles is 4-0 against Arizona this year, the first two wins coming in the season-opening series in Australia. ... Prado committed two errors and struck out three times. ... The Diamondbacks have struck out 23 times in the first two games of the series.