Greene scored the go-ahead goal in the third period and Los Angeles moved within a win of a playoff berth by defeating the Dallas Stars 3-1 on Saturday.
"How about that," Kings coach Terry Murray said. "He gave us a couple last year. We got used to it, and we want more."
Greene's second goal of the season -- his sixth in the NHL -- put the Kings on the verge of their second straight playoff appearance. Wayne Simmonds had a goal and an assist, Dustin Brown also scored, and Jonathan Quick stopped 24 shots for Los Angeles, 9-2-1 in its last 12.
"We just found a way to do it again," Kings forward Jarret Stoll said. "We found a way to get the points, and that's all that matters."
The Kings moved into a tie with idle Phoenix for fourth place in the Western Conference.
Brad Richards scored his 26th goal and Kari Lehtonen had 29 saves for Dallas, which continued its late-season collapse. The Stars have dropped six straight, and appear likely to miss the playoffs for a third straight year.
The Stars are 10th in the West, five points behind eighth-place Chicago with five games remaining.
"We're still alive here, and we're still kicking," Dallas coach Marc Crawford said. "You just have to keep believing that your luck is going to change.
"It was a very even game. They caught the break and we didn't. End of story."
Greene's one-timer from outside the circle beat Lehtonen stick side and snapped a tie 12:50 into the third. Simmonds sent a pass from behind the net, cleanly in front of the crease, to Greene for an uncontested shot that deflected in off the skate of Dallas defenseman Alex Goligoski.
Brown added an empty-net goal in the closing seconds for his team-leading 28th goal.
"We've been seeing that guys are stepping up, and we've been playing well without them," Simmonds said of injured top scorers Anze Kopitar and Justin Williams. "It would be nice to have them back, but we can't worry about that right now."
The Kings, held without a third-period shot in a 3-1 loss to Vancouver on Thursday, were aggressive from the start against Dallas.
Peppering the net with 12 shots in a scoreless first period, the Kings broke through early in the second on a deflection by Simmonds.
Willie Mitchell fired a shot from the point into heavy traffic, and the puck was deflected by Simmonds and dribbled behind Lehtonen. Stars defenseman Mark Fistric reached to clear the puck from the crease but knocked it in at 4:15.
"We put both goals into our own net, so that puck luck, or whatever you want to call it, isn't going our way," Crawford said. "All you can do is keep persevering. If we continue to work hard, then hopefully it will change."
The goal was Simmonds' 14th of the season, with assists coming from Mitchell and Trevor Lewis.
The Stars tied it 1-1 just over a minute later when they broke out on a 4-on-2 rush that ended with Jamie Benn and Goligoski assisting on Richards' slap-shot goal from the right circle.
Dallas had gotten as high as fourth in the West on March 13. A 3-2 loss to the Kings sent the Stars into a spiral, in which they have lost eight of nine.
The Stars were beaten 6-0 by San Jose on Thursday, and they are 0-3-1 on a five-game road trip that will conclude Sunday at Anaheim.
"We felt pretty confident as the game went on," Stars captain Brenden Morrow said. "It was a game we could get, we just didn't have that extra step to get to the loose pucks."
The Kings and Stars engaged in a relatively tame affair for the division rivals, committing just one penalty each. Dallas' Mike Ribeiro and Los Angeles' Brad Richardson picked up matching roughing penalties with 9 seconds left in the second period.
The teams combined for 91 penalty minutes in their last meeting in Los Angeles on Nov. 11.
The Kings are 2-1 without Kopitar (ankle) and Williams (shoulder). ... Los Angeles improved to 13-3-2 against Dallas since the beginning of the 2008-09 season. ... Lehtonen made his 19th straight start. Richardson had gone seven games without a goal. ... Los Angeles had no power plays in a game for the first time since March 31, 1984, against Edmonton.