Greinke (10-3) scattered six hits and issued one walk while striking out seven as the Dodgers improved to 15-4 in games started by the 2009 Cy Young Award winner.
Adrian Gonzalez bashed a two-run homer in the first inning while Nick Punto went 2-for-3 with a two-run single for National League West-leading Los Angeles, which clinched the three-game series and improved to 19-3 since the All-Star Game break.
Skip Schumaker supplied four hits for the winners.
Roberto Hernandez (6-12) was tagged for five runs on eight hits over four full innings for the Rays, who came into the contest two games behind AL-East-leading Boston.
Celebrity Families of 2014 [PHOTOS]