The teenager scored Dortmund’s third with a simple tap-in after goals from Axel Witsel and Raphael Guerreiro.
Guerreiro then made it 4-0 after capitalising on a poor back pass from Felipe Luis to finish past Jan Oblak.
The result was manager Diego Simeone’s biggest defeat in his seven years in charge of Atletico.
Witsel opened the scoring with a deflected shot in the first half but it was 73 minutes before Guerreiro made it two, finishing off a brilliant team move.
Former Manchester City winger Sancho came off the bench just after Dortmund’s second and got his name on the score sheet four minutes later when Mario Gotze played Achraf Hakimi through and he squared the ball for an easy finish.
The 18-year-old, who made his England debut this month as a substitute in a 0-0 draw against Croatia, became the second-youngest English player to score in the Champions League, after Alex Oxlade-Chamberlain for Arsenal against Olympiakos in 2011.
He is also the first English player to score for a non-British club in the Champions League since David Beckham for Real Madrid against Rosenborg in November 2005.
Atletico Madrid hit the post twice, either side of half-time, through Saul Niguez and Angel Correa but the German side were ultimately too much for them.
The result means Dortmund top Group A with nine points. Atletico remain in second, having won their first two games.