The 22-year-old initially looked to be struggling with the pace of the game as City dominated possession but failed to score.
But, collecting a chipped pass from Lucas Moura on the edge of the area, Bergwijn controlled with his chest before thumping the ball past Ederson for a remarkable opener in the second half.
"It's a dream for me that has come true," he told Sky Sports. "I'm so happy to be here and make a goal for Spurs."
Heung-min Son, who added Spurs' second, said of his new team-mate: "We are so happy that he's scored his first goal on his debut. Well deserved.
"Sometimes we had to defend all together but second half we came out strong. Steven scored an amazing goal and I think the lads did a really, really good job."