The England U17 players managed to cope with the searing heat and humidity in Kolkata, along with sustained pressure from the Brazilian side, to ensure they will be taking place in the final on Saturday.
Liverpool's Brewster was clinical throughout and was assisted by Manchester City midfielder Phil Foden as Steve Cooper's side hope to follow in the footsteps of the U20s, who won their World Cup back in June.
Brewster had his first goal after 10 minutes, firing past Brazao at the second time of asking, after an excellent delivery from Chelsea's 16-year-old midfielder Callum Hudson-Odoi.
Brazil then levelled the scoring after 21 minutes when Flamengo defender Wesley scored from close range after Manchester City's Curtis Anderson could only parry Paulinho's low effort.
However, just before half-time, Brewster put the Three Lions back in the driving seat as he pounced at the back post to put top of the tournament's goalscoring charts.
The 17-year-old striker completed his hat-trick and scored his seventh goal in five games on 76 minutes after a pin-point low cross from Foden, and England held on to withstand Brazil's late pressure.
England will face either Mali or Spain, who play at 15:30 BST, in Saturday's final.