Danny Chiasson scored in the third period for Saint John (29-12-2), while goaltender Mathieu Corbeil stopped 21 shots and turned aside four shooters in the shootout.
The Sea Dogs couldn't score on their two power plays, while Moncton had a goal on its only man advantage.
Both New Brunswick teams have struggled of late. The Wildcats have a single win in their last 10 games and Saint John has four victories in its past 10.
The Sea Dogs are still in second in the QMJHL despite their recent troubles.