Dillon connected with a wrist shot at 16:59 of the third to break a 3-3 tie and help the Stars to their fifth win in four games.
Dallas trailed 3-1 midway through the second period but rallied with three unanswered goals.
Cody Eakin had a goal and assist, and Reilly Smith and Antoine Roussel each scored for the Stars, who had lost eight of their last 10 meetings with the Canucks.
Richard Bachman took over goaltending duties after Kari Lehtonen was injured in the first period. Bachman stopped 18-of-21 shots.
Daniel Sedin, Chris Higgins and Alexandre Burrows scored for the Canucks, whose six-game winning streak came to an end. Henrik Sedin had two assists to become Vancouver's all-time scoring leader with 757 career points.