Joba Chamberlain struck out two in the eighth, and Joe Nathan struck out two in a perfect ninth for his 18th save in 23 chances.
Greinke (11-5) allowed three runs and seven hits in seven innings. He struck out eight without a walk.
''You look at their lineup today, and you think you are getting a break with Cabrera and Victor being out,'' Greinke said. ''But they were putting up great at-bats all day. I was throwing good pitches.''
Miguel Rojas of the Dodgers hit his first career homer in the fifth.
Scherzer escaped a first-inning jam thanks to some impressive fielding behind him. Yasiel Puig hit a one-out triple, but with the infield in, Hanley Ramirez hit a sharp grounder to Eugenio Suarez, and Detroit's rookie shortstop threw quickly to third to catch Puig off the base.
''That was a great play,'' Scherzer said. ''He just made an instinct play - saw Puig was off the base, and made a quick little snap throw. That just helped keep the game where it was.''
Rajai Davis made a sliding catch in left field for the third out.
Austin Jackson opened the bottom of the inning with a triple and scored on Ian Kinsler's single. Kinsler advanced to second on a wild pitch and scored on a two-out single by Don Kelly off Greinke.
''You let a guy like that settle in, it can be a long day,'' Jackson said. ''For us to kind of get something going there that first inning, that definitely sparked us.''
Greinke was sharp after that. Detroit scored again in the fourth on an RBI single by Nick Castellanos, but the Los Angeles right-hander retired the last nine hitters he faced, striking out the final four.
But the Dodgers couldn't overcome Scherzer. They put men on first and second in the third, but Puig hit into a double play. In the sixth, Puig hit a leadoff double, but the next three batters couldn't hit the ball out of the infield.
''I like where I'm at,'' said Scherzer, the reigning American League Cy Young Award winner. ''I feel like my curveball is improved from where it was at the beginning of the year. I feel like it's more consistent from start to start. At this point in time, I feel like I'm a better pitcher than I was in April.''
Matt Kemp went hitless with three strikeouts. He's now 0 for 21 against Scherzer.
Torii Hunter's sacrifice fly in the eighth gave Detroit another run.
Dodgers manager Don Mattingly doesn't plan any major changes to the lineup when OF Carl Crawford returns from a sprained left ankle. There could be a logjam with OFs Kemp, Puig and Andre Ethier also in the fold. ''We're playing pretty well,'' Mattingly said. ''We'll just talk about it with (Crawford) - let him know where everything is.'' ... Kelly drew the second intentional walk of his career in the eighth inning. ... Detroit plays at Kansas City on Thursday night. The Tigers send LHP Drew Smyly (4-8) to the mound against RHP Jeremy Guthrie (5-7). ... The Dodgers return home to take on San Diego. Los Angeles ace Clayton Kershaw (10-2) faces RHP Odrisamer Despaigne (2-0) of the Padres.