The Belgian notched his first goal in the Champions League with a well-timed volley in the 83rd minute after he had set up forward Son for a 47th minute lead. Substitute Fernando Llorente grabbed their third in the 86th.
That was Son’s ninth goal against Dortmund in 11 matches, with the South Korean, who played in the Bundesliga from 2010-15, having also scored twice in the competition’s group stage last season.
The Bundesliga leaders, who were eliminated from the German Cup last week, were nowhere to be seen in the second half, with speedy teenager Jadon Sancho given no space to operate after the break. The return leg is on March 5.
Before Wednesday, Tottenham, playing without injured captain Harry Kane and Dele Alli, had never won a home match in the knockout stage of the competition and Lucas Moura came close with a fine shot in the seventh minute.
Dortmund, who were themselves missing injured captain Marco Reus and top striker Paco Alcacer, gradually took control and had the better chances in the first half with Christian Pulisic firing onto keeper Hugo Lloris and the Frenchman saving a point-blank header from Dan-Axel Zagadou.
But the hosts changed the course of the game two minutes after the restart, pouncing on a defensive error and with Vertonghen whipping in a cross for Son to tap in.
Dortmund, who had conceded just two goals in the group stage, tried to recover but found no way past the disciplined Tottenham backline.
Vertonghen then timed his run to perfection in the 83rd to kill off the game and Llorente gave them a big advantage for the second leg.