Son converted Spurs' first shot on target at Wembley on Wednesday as the Bundesliga leaders were made to rue a string of first-half misses.
Mauricio Pochettino's side struggled in the opening 45 minutes but, as has often been the case in the Champions League this season, they found the net in the second period.
After Son's first European goal of the campaign, late efforts from Vertonghen and Llorente put the Premier League title contenders in a fantastic position heading to Signal Iduna Park in three weeks.
Tottenham went agonisingly close to opening the scoring in the seventh minute when Lucas Moura's volley fizzed wide of goal, the Brazilian having controlled Davinson Sanchez's header on his thigh.
It would be as close as Spurs would come, though, as Christian Pulisic, Axel Witsel and Thomas Delaney all forced saves out of Hugo Lloris.
The France international had to be at his acrobatic best in the closing minutes of the half as Dan-Axel Zagadou rose to meet Jadon Sancho's cross, the goalkeeper clawing the ball away at full stretch.
Having weathered the storm prior to the break, Spurs opened the scoring two minutes after the restart as Son peeled away from Zagadou to volley Vertonghen's pinpoint cross past Roman Burki.
Toby Alderweireld went close to doubling Tottenham's advantage from a Christian Eriksen corner moments later, but Burki was on hand to keep his header out.
Mahmoud Dahoud wasted a great chance to draw Dortmund level, his meek effort easily saved, and Vertonghen showed him how it should be with a fine volley from Serge Aurier's deep cross.
Substitute Llorente headed in a third in the closing minutes from Eriksen's corner to ensure Spurs take a commanding lead into the return fixture.
2013 - Jan Vertonghen has both scored and assisted in the same game for just the second time in his Tottenham career, last doing so in March 2013 against Swansea in the Premier League. Surge. #TOTBVB— OptaJoe (@OptaJoe) February 13, 2019
What does it mean? Advantage Spurs
While he will be delighted his side emphatically won the game, keeping a clean sheet will be just as pleasing for Pochettino. Stopping Dortmund registering an away goal is a major achievement and sets Spurs up nicely for their trip to Germany.
Son is Spurs' lucky charm
Forget Harry Kane, forget Dele Alli, Son is Tottenham's lucky charm at the moment. For a 13th time this season, Spurs have gone on to win when he has scored. He had a quiet opening 45 minutes but pounced when presented with his first chance.
Zagadou goes missing
Moments before half-time, Zagadou could have been Dortmund's hero. Instead, early in the second half, he became the villain. As Vertonghen shaped to cross, the teenage defender allowed Son to find space and set Spurs on their way to a handsome victory on home turf.
Key Opta facts
- This was Tottenham's joint-biggest Champions League victory, winning by a three-goal margin for the fifth time in the competition.
- Spurs are the first English side to win a home Champions League knockout game against German opponents since Manchester United defeated Schalke in the 2010-11 semi-final.
- Llorente scored just two minutes and 26 seconds after coming off the bench for Spurs.
- Son has scored nine goals in 11 games in all competitions against Borussia Dortmund - more than he has against any other side in his senior club career.
- For the fifth time in their history, Borussia Dortmund were beaten 3-0 in a Champions League game. The result equals their biggest defeat in the competition.
Tottenham travel to Burnley in the Premier League on Saturday, with clashes against London rivals Chelsea and Arsenal also to come before the return leg in Europe. Dortmund also have three league matches between now and March 5, as they go up against Nurnberg, Augsburg and Bayer Leverkusen.