Simmonds had a goal and an assist, and Jonathan Quick made 22 saves in the Kings' 3-1 victory over the slumping Edmonton Oilers on Wednesday night.
"We had two games at home that we lost where we didn't work as hard as we should have, so that was our goal tonight, to come in and play a solid 60 minutes," Simmonds said. "I think we did that and it was a big character win for us."
Greene, a former Oilers player acquired by the Kings in the trade for defenseman Lubomir Visnovsky, gave the Kings a 2-1 lead with 4:28 left. Greene beat Jeff Deslauriers with a point shot high to the stick side.
"That was a big one," Greene said. "Mind you, any goal for me is a big one. They don't come too often. It was a good feeling to get a game-winner against my old team. I was just trying to shoot it hard. I had no idea where it was going."
Frolov iced the game with an empty-net goal.
The Oilers have won four of their last 16 games to drop to 10-12-3.
"It was one of those games that was up there for grabs by one of the two teams and they got it and we didn't," Oilers coach Pat Quinn said. "We played just poorly enough to not win it and they played just well enough to win it. They're going home with two points and we are sucking wind."
Edmonton took a major blow in the second period when Ales Hemsky was drilled into the boards by Michal Handzus. Hemsky sustained an upper-body injury and didn't return.
"He got hit earlier in the game as well but this one was full force into the boards," Quinn said. "It was a hard hit and he got himself hurt. I don't know how long he will be out. But it's not real positive right now."
It was the first meeting between the two teams this season. Last season, they split four one-goal games. ... Los Angeles scoring star Anze Kopitar has gone seven games without a goal.
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EDMONTON, CANADA - NOVEMBER 25: The vacant dressing room of the Edmonton Oilers waits before a game between the...