The striker opened the scoring in the Swans' first Championship match in over three months after 18 minutes of play, converting a cross from close range.
The Liverpool loanee then grabbed his second of the afternoon moments later with a well-taken volley to put Steve Cooper's side 2-0 up at the Riverside Stadium.
Those strikes took the 20-year-old's goal tally for Swansea to six since completing a temporary switch in January.
Reacting to Brewster's display, Swansea boss Cooper said: "Rhian looked a real threat, got his two goals and he's disappointed he didn't have three.
“He's a really good player but we are a team that likes to play as a team.
“Rhian obviously will get some headlines because he got a brace but it's as a consequence of the team playing the way we want to play.”
Elsewhere in the division, Ovie Ejaria started Reading's 1-1 draw with Stoke City, while Herbie Kane's 90-minute outing for Hull City ended in a 1-0 defeat to Charlton Athletic.
And over in Germany, Marko Grujic played the entirety of Hertha BSC's 2-0 triumph over Bayer Leverkusen in the Bundesliga.