With the score even at 3-all, LaPerriere worked a 2-on-1 with Paul Stastny, converting his return pass at 3:58.
Antti Laaksonen's power-play goal at 13:24 of the final period stretched Colorado's advantage to 5-3.
Milan Hejduk, Tyler Arnason and Andrew Brunette added goals for the Avalanche (4-1).
Jussi Jokinen, Jaroslav Modry and Mike Modano scored for the Stars (2-4).
Jose Theodore was Colorado's starting goaltender, allowing three goals on six shots before he was replaced by Peter Budaj at 10:43 of the second period. Budaj stopped all nine shots he faced.
Mike Smith, who has won the backup goaltender's job behind Dallas starter Marty Turco, played the entire game and had 25 saves.
Dallas took a 2-0 lead in the first period, scoring on its first two shots. Jokinen tallied his third goal of the preseason at 1:23 and Modry added a power-play goal at 9:16.
Colorado came back with three straight goals to take the lead. Hejduk made it 2-1 at 9:47. Arnason and Brunette followed with power-play goals to give the Avalanche a 3-2 lead.
Modano capped a wild first period with a power-play goal at 19:18 to tie it at 3.