Brendan Hurley scored the lone goal in regulation for the Hitmen (12-4-3), who have won four straight games.
Carter Hansen scored for the Warriors (7-7-6) at the 1:51 mark of the second period before Hurley replied for the Hitmen 15 seconds later.
Moose Jaw has now lost six straight games, although two of those were in a shootout and two others in overtime.
The Warriors outshot the Hitmen 29-19, including 5-2 in overtime.
Brooks Macek of the Hitmen and Justin Kirsch of the Warriors scored in the first round of the shootout before Chase had the only other goal in the third round to give the Hitmen the victory.
Hitmen goalie Chris Driedger made 28 saves through regulation and overtime before stopping two of the three shots he faced in the shootout.
Justin Paulic made 17 saves in the Moose Jaw net in a losing cause.