Greinke fell to 12-7 with a third loss in his last four decisions, even though two of the five runs charged were unearned because of the first of two Hanley Ramirez throwing errors.
"It's a playoff-type game and that's a playoff-type team and if you make mistakes, you pay, simple as that," said Dodgers manager Don Mattingly.
Greinke was outpitched by a dominating Garrett Richards (12-4), who allowed five singles and struck out nine, joining Aaron Sele as the only Angels starters to shut out the Dodgers.
Greinke went seven innings, giving a breather to a bullpen that had pitched 14 2/3 innings in the three-game Cubs series over the weekend. The last two scoreless innings were pitched by Carlos Frias, promoted before the game and making his Major League debut.
In the duel between center-field sensations, Mike Trout of the Angels went 1-for-4 with an RBI double, run scored, strikeout and steady boos. Trout also lost a double on a leaping catch by third baseman Juan Uribe, with many of the outs for Greinke coming on hard-hit balls.
Yasiel Puig went 0-for-4 with two strikeouts and one finger wag to show up Erick Aybar when Puig's throw from center field just missed doubling Aybar off first base. Puig's hot-dogging later turned embarrassing, as the slow-footed Albert Pujols tagged and advanced to second when a lackadaisical Puig caught Josh Hamilton's high fly and assumed no throw was needed.
Puig waved for Pujols to tag to third when he caught the next fly ball. At the end of the inning, Puig and Pujols exchanged words; Uribe and Mattingly then spoke to Puig in the dugout.
"I hope he learns a lesson from it," said Mattingly. "Albert knows how to play the game. We talked about them being an aggressive team. If you don't pay attention, they will embarrass you, and he embarrassed him there."
The first inning was a whole other kind of embarrassment for the Dodgers: two missed diving-catch attempts by left fielder Carl Crawford, two wild pitches by Greinke, a throwing error by Ramirez and second baseman Dee Gordon was flattened in a collision with second-base umpire Jim Reynolds.
"A good baseball man told me the two most important innings are the first and the ninth," Mattingly said. "The first inning can cost you. The very first play of the game can cost you."
After being ambushed in a four-run first inning, Greinke didn't allow another hit until Josh Hamilton homered on a change-up leading off the sixth.
"Probably up there with the worst pitches I've ever thrown. I couldn't believe I threw that kind of pitch," Greinke said.
This was the third loss in the last four games for the Dodgers, who are stumped over what to do with rotation back-enders Dan Haren (five-start losing streak) and Josh Beckett (hip impingement). Part of the club's strategy is to hope that Greinke, Clayton Kershaw and Hyun-Jin Ryu keep winning.
But the offense isn't doing its part either, scoring 16 runs total in the last six games. Their only real chance against Richards came in the second inning after a leadoff single by Ramirez and a four-pitch walk to Matt Kemp. But Crawford struck out, Uribe's foul fly was run down by Hamilton in the left-field corner, A.J. Ellis walked and Greinke flied out.
"It's just unbelievable what he's doing on the mound this year," Pujols said of Richards. "He should be an All-Star. I'm pretty sure he's going to have many opportunities for that, but he's been outstanding for us."
Crawford tried to make two diving catches on the first two batters of the game and went 0-for-2. He trapped Cole Calhoun's sinker for a single, then totally whiffed on Trout's drive that rolled to the wall for an RBI double only because Trout slipped or would have had a triple. Pujols followed with an RBI double to the left-center gap.
"I wasn't even close to hitting the spot to Trout and Pujols and they did what they should do," said Greinke. "By then it was too late."
One out later, Adrian Gonzalez saved a run by throwing Pujols out at the plate after fielding Aybar's grounder. But Aybar took second on Greinke's wild pitch and scored when Ramirez threw away Howie Kendrick's grounder. Kendrick advanced to second on another wild pitch and was singled home by David Freese.