Colorado beat Dallas for the 11th time in 12 meetings during Lindy Ruff’s three seasons as Stars coach.
Landeskog's goal at 10:12 of the second period broke a 1-all tie. Tyson Barrie pushed the puck ahead to Landeskog, who skated in alone and put a wrist shot past Niemi.
Colorado's penalty killers are 19 for 21 in the last seven games, with three short-handed goals.
Demers scored for Dallas on the power play at 7:24 of the second.
Midway through the first period, Niemi saved a close-in shot by Soderberg with his pads and engulfed John Mitchell's wrist shot from the slot.
Soderberg gained redemption during the first period's only power play. He won a battle for the puck from Jamie Benn, skated in on Niemi and flipped a wrist shot into the upper left corner at 19:07.
Dallas outshot Colorado 10-7 in the first.
In the second period, the Stars had the first 12 shots on goal, finally scoring on the ninth when Demers backhanded in a rebound from the blue paint.
The Stars' goal appeared to come with a bonus. Soderberg hit Demers with a high stick that drew blood and a double-minor.
Dallas outshot Colorado 19-4 in the period, but still came away trailing by a goal.
Grigorenko put in his own rebound over Niemi's glove for a 3-1 lead.
NOTES: Avalanche RW Jack Skille left in the first period with a back injury after crashing into the boards following a shot that Niemi saved. . Colorado welcomed back D Erik Johnson, who missed nine games because of a lower-body injury, and LW Blake Comeau, who missed Friday's game while his wife was giving birth to their second daughter. . The short-handed goals moved the Avalanche up to fifth in the NHL with six, trailing Ottawa (nine), Montreal (eight) and Dallas and Winnipeg (seven each). . Landeskog has three goals in the two games vs. Dallas this season. ... The Stars are 1-8-3 against Colorado in Ruff's tenure as coach, including losses in both games this season. ... The Stars scored once on five power plays, but gave up the two short-handed goals and are 4 for 33 over the last 12 games.