Steve Cooper’s side trailed in the early stages of the Championship clash but Brewster – making his second appearance – brought them level on 19 minutes with a clinical finish.
Collecting a pass into the area with an angled run, the 19-year-old swept an emphatic low shot across the goalkeeper and into the bottom corner.
Swansea went on to claim victory, boosting their push for promotion, as Andre Ayew struck a second for the home side after the interval.
“What a great time to get [my first goal],” said Brewster. “It was great to get a goal, hopefully it’s the first of many.