The win gives Los Angeles 96 points on the season and the fourth spot in the Western Conference, three back of San Jose for the top spot in the Pacific Division.
Los Angeles' next game is in San Jose on Monday.
Wayne Simmonds had a goal and an assist while Jonathan Quick stopped 24 shots for the Kings, who have won five of six.
"This was an important game and like the month of March, every game that we've be involved in it's been a playoff-like attitude compete," said Los Angeles head coach Terry Murray. "It was a great win by the group of guys in that locker room today."
Brad Richards had the lone goal while Kari Lehtonen stopped 29-of-31 shots for the Stars, who have dropped six straight and eight of nine.
"We're still alive and kicking and you just have to believe that your luck is going to change," said Dallas head coach Marc Crawford. "It was a very even game and they caught the break and we didn't, end of story."
The loss is a big blow to any hopes of a playoff spot for Dallas. The Stars are five points back of Chicago with five games to play.
With the score tied in the third, Greene blasted a shot from the top of the right circle and it hit off the skate of Mark Fistric and went into the net for a 2-1 lead.
Dallas pulled Lehtonen to try for the equalizer, but Dustin Brown sealed the win with an empty-netter in the final moments.
After a scoreless first period that saw each goaltender make 12 saves, the Kings took a 1-0 lead at the 4:15 mark of the middle session.
Willie Mitchell blasted a shot on net from the right point that hit off of Simmonds and was slowed down by the goaltender. It was headed into the net and Fistric tried to clear it out, but he couldn't get enough on it and the puck went in.
However, Dallas tied the game 1:13 later as a wrister from Richards on the fly at the right circle trickled through Quick for his 26th goal of the year.
Dallas plays at Anaheim on Sunday...The Kings took four of the six games against the Stars this season...Neither team had a power play in the game as the only penalties were a pair of matching minors to Mike Ribeiro of Dallas and Brad Richardson on Los Angeles late in the second period.
For more news and information on today's game, be sure to read the SB Nation blog Jewels From The Crown.