Then Cueto and his Giants teammates took command to pave the way to a 4-2 victory in the opener of a four-game series.
"Johnny just did a great job," Giants manager Bruce Bochy said. "He lost some pitches early, but he just settled down to pitch a great game. It was a good one for him and us."
Cueto gave up two runs on eight hits in seven innings to improve to 7-2 against Arizona. He struck out nine and threw 115 pitches. As for Greinke, the right-hander was charged with four runs on eight hits in six innings. He walked three and struck out four in the 104-pitch effort, falling to 8-1 against the Giants.
The D-backs scored in the first inning on a run-scoring single by Jake Lamb for a 1-0 lead. The pitchers then traded zeros in the second and third frame, but the Giants jumped ahead with two runs in the fourth. A two-run home run by Joe Panik extended the visitors' lead to 4-1.
The D-backs answered in the sixth on a RBI triple by Chris Owings to cut the Giants' lead in half, 4-2. Cueto struck out Nick Ahmed on three pitches with runners on the corners to end the scoring threat.
"They had some really good at-bats, I felt like," Greinke said. "They took a lot of close pitches and were ready to hit anytime I made a mistake. I thought a couple of their guys had some really good at-bats."
MOMENTS THAT MATTERED The mighty Zack: Greinke came into the game with an 8-0 record and 2.12 ERA in 11 career starts against the Giants, who pounded him for four runs on seven hits in the fourth and fifth. The big blow was Panik's first-pitch two-run homer with none out in the fifth. The Giants had eight baserunners against Greinke in those two innings. More >
Lopez to the rescue: After closer Santiago Casilla loaded the bases with two outs in the ninth, Bochy turned to Javier Lopez, who induced a groundout by Lamb to earn his first save of the season. More >
Jean (hit) Machine:Jean Segura extended his hitting streak to 10 games with a single in the first and has reached base in a career-high 19 games. He scored the D-backs' first run.
The leadoff hitter remains among the National League leaders in games with three or more hits, hits and singles. He finished 3-for-5 and is hitting .353.
"The competition is stiff in this league. Every time you go somewhere or they come to your place, there are good pitchers pitching and good players. We just have to outplay them. We have not done that here." -- D-backs manager Chip Hale, on the division and the club's 5-13 home record
WHAT'S NEXT Giants: It'll be Jeff Samardzija in the second game of a four-game set Friday, with a 6:40 p.m. PT start. Bochy said injured left fielder Angel Pagan will play for the first time since suffering a Grade 1 left hamstring strain May 1 in New York. He had missed the past 11 games.
D-backs: Shelby Miller had his best outing of the year in his last start against the Braves, and the club hopes he can continue the trend at home against the Giants on Friday at 6:40 p.m. MT. Miller has an 8.24 ERA at Chase Field this season.