"He's a great pitcher," Montero said. "He's a tough guy to face because all his pitches are nasty. Every single pitch he has is a plus."
Juan Uribe's ninth-inning home run sent the game into extras after Dodgers reliever Chris Withrow wild-pitched a run home in the top of the ninth during an intentional walk to Martin Prado.
The Dodgers were held to just Uribe's blow plus Scott Van Slyke's solo shot off Wade Miley leading off the seventh, which got Greinke off the hook. The Dodgers have scored just seven runs in their past four games.
"I thought all our pitching was good," manager Don Mattingly said, noting that his bullpen gave him six innings and allowed just the Withrow run, before Aaron Hill touched Chris Perez for a two-run single in the 12th. "We had plenty of time to score. [Offensively] we didn't do enough to win the game."
The D-backs had lost their first five games against the Dodgers this season.
"Our pitchers did really well, keeping us in the game," catcher Tim Federowicz said, "but we weren't able to get a big hit after Juan kept us going."
The start was Greinke's 16th consecutive of at least five innings and allowing two earned runs or fewer -- the longest such streak in the Major Leagues since 1914. He is 10-1 with 1.76 ERA in that span.
"I think it takes a lot of skill to do it, but you have to get a lot of breaks," Greinke said of his achievement. "I've had more than my share."
He gave up a leadoff hit to Gerardo Parra in the first, got a double-play ball, then uncharacteristically walked the next two hitters. Greinke hadn't walked a batter in his past two starts.
"They were kind of patient," Greinke said of the D-backs, "but in the first three innings, I didn't throw enough strikes. I wasn't executing early on. It just comes sometimes. Each inning, it got a little bit better."
But the pitches added up to 105 through six when Mattingly decided to take Greinke out. Asked if he considered leaving his starter in for the seventh, especially on a night when he was resting relievers Kenley Jansen and Brian Wilson, Mattingly said: "Not when he climbed over 100, not at this point [of the season]."
A 12-pitch at-bat by Montero in the fourth before he grounded out preceeded the D-backs catcher's second career homer off Greinke, two innings later.
"It didn't come to my mind until you just mentioned it," Greinke said of his last two battles with Montero, who homered on a 2-1 curve. "The main thing is, you just want to make good pitches when you fall behind. I threw a quality pitch, and he made a really good swing."
Montero has six hits in 18 at-bats against Greinke.
Van Slyke's homer, his second off Miley this season, was his fifth hit in 13 at-bats against the Arizona left-hander. It was also Miley's exit cue.
He went six-plus innings, struck out eight and walked five in a no-decision after losing twice to the Dodgers earlier, but Miley's 3-0 career mark at Dodger Stadium remained intact.
"He had better command tonight, he attacked more, threw a lot more strikes early," Mattingly said. "The two games before, we were getting into favorable counts."