After all, the last time he faced them, the game erupted into a bench-clearing brawl that resulted in eight suspensions.
"I might want to win a little more," he said. "That's also because they're in first place, so it's a big game because of that."
Whatever the reason, Greinke dominated the Diamondbacks, giving up two hits in seven scoreless innings and the Los Angeles Dodgers snapped Arizona's five-game winning streak with a 6-1 victory.
"This is probably maybe the second time he's had this kind of stuff," Dodgers manager Don Mattingly said. "Hopefully we're trending toward this more consistently."
Greinke (7-2) struck out seven and walked two. He also had a career-high three hits.
"That is how he won a Cy Young right there," said Arizona's Jason Kubel, who with Minnesota saw Greinke a lot when the pitcher was with Kansas City.
"He has great stuff and he swung the bat well, too. He throws hard. He throws that hard slider and it is hard to pick up until it is too late and his location is big for him. He had it going."
Greinke has won four in a row but he hasn't exactly been electric recently. In his previous start against Colorado, he allowed five runs and six hits and walked seven in six innings. That prompted some work to alter his mechanics.
"It's got to be more consistent," he said. "Today was good but it can't just be one game and then a couple of OK games and then a bad game and then a good game."
Greinke was 0-2 with an 8.71 ERA in four previous starts at Chase Field.
"I've pitched really bad here," he said, "some of my worst games ever at this place, so that was nice."
Hanley Ramirez singled three times to extend his career-best hitting streak to 19 games and was one of five Dodgers with an RBI.
Los Angeles, winners of 13 of 16, tied its season high with 17 hits -- 16 of them singles. Andre Ethier had three hits and A.J. Ellis drove in two runs as the Dodgers pulled within 3 1/2 games of the first-place Diamondbacks in the NL West.
"Without this little streak we've been on who knows what position we'd be in," Ellis said. "We also got a little help from the NL West coming back to the pack. Nobody ran away with it."
Randall Delgado (1-3) allowed three runs and 11 hits in six innings to take the loss.
Arizona averted the shutout on Eric Chavez's RBI single in the ninth.
It was the opener of the first series between the teams since the three-game set that featured the June 11 melee in Los Angeles, resulting in eight suspensions.
Greinke was a major player in the events leading up to the incident. Ian Kennedy, who hit Yasiel Puig and Greinke with high pitches to trigger the melee and served a 10-game suspension, starts for Arizona in the second game of the series on Tuesday night.
Puig and Greinke were booed roundly when they came to the plate.
Greinke struck out four of the first five and didn't allow a hit until Paul Goldschmidt's leadoff single in the fourth. The only other hit off the Los Angeles right-hander was Martin Prado's two-out double in the seventh. Goldschmidt also had an infield single in the ninth and scored Arizona's run.
Ramirez and Ethier singled with one out in the fourth. A.J. Ellis flew out to center, then Juan Uribe singled Ramirez home from second to put Los Angeles up 1-0. Ethier was stranded at third when Mark Ellis flew out to center.
Greinke's single started a two-run fifth that put the Dodgers up 3-0. After one out, Puig singled, then Adrian Gonzalez and Ramirez each got an RBI hit to right. Kubel mishandled Ramirez's hit but threw the runner out trying to take second on the play.
The Dodgers scored two off reliever Charles Brewer with two outs in the sixth on Ethier's RBI double and Ellis' run-scoring single.
Arizona center fielder Gerardo Parra got a big ovation after chasing down Ellis' fly with a lunging grab as he fell to the warning track to start the sixth.
Before the game, the Dodgers placed Matt Kemp on the 15-day disabled list, retroactive to Saturday, with joint inflammation in his left shoulder. Kemp is to see the team doctor in Los Angeles on Tuesday.
Ricky Nolasco, acquired from Miami on Saturday, makes his first start for the Dodgers on Tuesday night. He took the roster spot of RH Stephen Fife, who was placed on the 15-day DL for the second time this season with right shoulder bursitis. ... Ramirez's hitting streak is the longest active one in the majors and longest by a Dodger since Ethier's 30-game streak from April 2 to May 6, 2011. ... Outfielder Scott Van Slyke was recalled from Triple-A Albuquerque to take Kemp's roster spot. ... The Diamondbacks announced the signing of six international players, including 16-year-old Venezuelan catching prospect Jose Herrera. The other five signees also are 16. ... Greinke had allowed a combined nine earned runs and 18 hits in his previous two starts, although he won both.