"He put a heck of a swing on a curveball there and got himself an RBI knock," Arrieta said. "I wanted that pitch below the zone. It was below the zone, but just low enough. Greinke's a great athlete and he found a way to get it into left field. He can swing the bat a little bit."
After giving up a single to Starlin Castro to open the game, Greinke retired the next 13 batters before Murphy singled in the fifth.
Greinke got the first two outs of the ninth before running into trouble. Anthony Rizzo doubled, then Greinke hit Nate Schierholtz before Bogusevic's RBI double.
Greinke issued consecutive one-out walks to Murphy and Welington Castillo in the eighth before Darwin Barney grounded into a fielder's choice to third. Murphy advanced to third and Castillo was out at second.
The Dodgers thought they had completed the double play, but second baseman Mark Ellis' throw didn't beat Barney to first, leading Dodgers manager Don Mattingly to shout, "Out!" from the dugout.
Greinke then got pinch-hitter Dioner Navarro to ground out to third, ending the inning.
Yasiel Puig hit his 13th homer on his first pitch from Michael Bowden with two outs in the eighth.
Hanley Ramirez homered off Carlos Villanueva with one out in the seventh, extending the Dodgers' lead to 5-0.
Los Angeles had tacked on a pair of runs in the sixth on Nick Punto's RBI double and Carl Crawford's sacrifice fly that made it 4-0.
The Dodgers took a 2-0 lead in the fourth on two-out hits by Greinke and Crawford. Greinke's single scored A.J. Ellis, who walked, and Crawford's ground-rule double to deep left scored Punto, who walked.
Notes: Arrieta fell to 0-2 against the Dodgers this season, having lost to them on April 21 when he was with Baltimore. ... The Cubs fell to 30-35 on the road, where they are better than at home (25-41).