Last year's finalists, who lost to Real Madrid at Wembley in June, got off to a fast start in the newlook league stage.
New Dortmund coach Nuri Sahin brought Gittens on as a substitute with 22 minutes remaining and the game goalless.
The 20-year-old scored shortly after, his right-foot shot looping over the head of Club Brugge goalkeeper Simon Mignolet with the aid of a deflection.
Again coming off the left flank, Gittens blasted a low shot into the net to score his second with four minutes remaining, snuffing out Brugge's chance of a comeback.
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