Green got a stick on a shot by Wayne Simmonds that went off the skate of Dallas defenseman Alex Goligoski and poked it past Kari Lehtonen to give Los Angeles a key win in the tightly packed Western Conference.
Simmonds scored in the first period and Dustin Brown added an empty-netter for the Kings, who with 96 points established a four-point playoff cushion in the West with four games to play.
Jonathan Quick stopped 24-of-25 Dallas shots to earn the win.
Brad Richards found the net for the Stars, whose dwindling playoff hopes took a possibly fatal blow with the setback. Dallas came into the game five points out of a post-season berth.
Lehtonen finished with 29 saves.
Notable deaths of 2014 [PHOTOS]