Sceviour's second goal at 9:44 gave the Stars a 4-0 lead and prompted Colorado coach Patrick Roy to pull goalie Jean-Sebastian Giguere in favor of Sami Aittokallio.
Gabriel Landeskog answered for Colorado with 2:11 left in the second period, and Brad Malone and Paul Stastny followed with goals in the third period to get the Avalanche within 4-3.
Sceviour added an empty-net goal with 1:40 left in the game. Kari Lehtonen stopped 26 shots for the Stars.
Giguere made six saves and Aittokallio stopped all 15 shots he faced.
"We had a really good start, but unfortunately for us, coming [in the first intermission] at 0-0 was probably the killer," Avalanche coach Patrick said, "but overall the guys played really well in that first period. We saw a lot of good things out there. We kept [Dallas] to one shot on goal in the first [period], and I thought that was great."