Spurs trailed 4-3 on aggregate in the quarter-final tie when Fernando Llorente grabbed a vital goal off a corner kick in the 73rd minute.
City appealed for handball but after a VAR review, it was determined the contact off the Spaniard's arm was incidental and the ball went in off his thigh.
With Llorente's strike standing, City needed to find another goal to reach the semi-final and they thought they'd done just that when Raheem Sterling scored a dramatic winner in stoppage time.
But VAR again came to Tottenham's rescue as a review deemed Sergio Aguero to be offside in the build-up to the England international's goal.
Son was well aware that video review technology very much worked in his side's favour on Wednesday night.
"I have never seen something like this," the South Korean told BT Sport. "It was tough and crazy but we are very proud of our team-mates. It was madness.
"Sometimes you are annoyed with VAR but today it is thank you. We fought together for 90 minutes. We showed unbelievable character and fight."
Son scored two goals on the night, becoming the top scoring Asian in Champions League history.
The 26-year-old will look to add to that record in the semi-final, where Tottenham will take on Ajax.
Spurs have now reached the last four in Europe for just the second time in the club's history, having last done so in 1961-62 under Bill Nicholson.
The club are also the seventh from England to reach the semi-finals of the Champions League, joining Manchester United, Man City, Liverpool, Chelsea, Arsenal and Leeds to help England overtake Spain as the country to have produced the most unique semi-finalists.
Liverpool and Barcelona will contest the other semi-final, with the winners meeting in the final on June 1 at the Wanda Metropolitano Stadium in Madrid.