Marchment, who signed a four-year contract with the Stars ($4.5 million AAV) on July 13, also scored at 2:35 of the second period to give Dallas a 3-2 lead.
Joel Kiviranta had a goal and an assist, and Radek Faksa had two assists for the Stars. Scott Wedgewood made 10 saves.
Shayne Gostisbehere, Nick Bjugstad and Vladislav Kolyachonok scored for the Coyotes. Ivan Prosvetov allowed three goals on 20 shots, and Jon Gillies made four saves in relief.
Kiviranta assisted on Ryan Shea's goal at 14:18 of the first period that gave Dallas a 1-0 lead, then put them back in front when he scored shorthanded 1:22 into the second to make it 2-1.
Gostisbehere's power-play goal with five seconds left in the first tied it 1-1, and Bjugstad scored with the man-advantage at 1:59 of the second to make it 2-2.
Kolyachonok tied it 3-3 with a shorthanded goal at 11:50 of the third period.
