Mauricio Pochettino’s side were on course for the last eight after Son’s first-half goal put them 1-0 up on the night and 3-2 ahead on aggregate.
But the Italian champions stunned Wembley with two goals in three second-half minutes from Gonzalo Higuain and Paulo Dybala.
In the dying seconds Harry Kane thought he had taken the tie to extra time, only to see his header come back off the inside of the far post before being cleared off the line.
Son, who was in tears after the final whistle, said: “That hurt, but it’s football. We played well but sometimes football is like this. We have to accept the result but still for me, it hurts a lot.
“What was the difference? Nothing. At the start we played well, we created lots of chances, got to 1-0 and made the perfect start I think.
“This game we can learn something from, but this is sad news. The dressing room was quiet.”
Tottenham’s European campaign, including wins over Real Madrid and Borussia Dortmund, was a vast improvement on the previous season, when they failed to get past the group stage.
South Korea’s Son said Spurs were “unlucky” to be out of the Champions League but had earned valuable experience.
“We remember last season, we went out in the group stage so now we did well, we learned something again and next season we will do better,” he said.
“Real Madrid and Borussia Dortmund doesn’t matter anymore. I thought we were better than (Juventus) but it doesn’t matter now — they scored two goals at Wembley Stadium,” he added.
And Pochettino insisted Tottenham had nothing to be ashamed of after three minutes of madness ended their European run.
Tottenham were on course to reach the quarterfinals for the first time since 2011 until Juventus completely flipped the script. The result was a bitter blow for Spurs and the loss will raise fresh questions about their ability to win major trophies under Pochettino.
But, yet to land any silverware with the north Londoners, Pochettino bristled at suggestions Tottenham aren’t mentally strong enough to win on the big occasions.
“All this talk is rubbish. At the end it is win or lose. When you assess the game, I think only Tottenham was better for more than 70 minutes,” he said.
“If Harry (Kane) scored at the end or we scored twice in the first half, maybe we are talking differently. I’m happy with my players. We did everything to try to win.
“It was not a lack of experience, not a lack of concentration. How many chances did we concede?
“We conceded three chances and they scored twice. We had a lot of chances and only scored one.
“We can talk about a lot of situations, but sometimes you need some luck to win.”