Spurs were 3-0 down on aggregate in their semi-final after goals from captain Matthijs de Ligt and Hakim Ziyech put Ajax 2-0 up on the night in Amsterdam.
But two goals before the hour from Lucas Moura put the tie back in the balance and the Brazilian forward capped a remarkable encounter with another fine finish in the sixth minute of stoppage-time.
It means an all-English final in Madrid on June 1 after Liverpool completed their own improbable feat of escapology from 3-0 down to Barcelona.
But, while Jurgen Klopp's side have enjoyed a stunning season that could still see them claim the Premier League title this weekend, Spurs failed to sign any new players and left Pochettino grappling with continuous injury problems as the move into their new stadium was long delayed.
"To get the club to the final of the Champions League is very close to a miracle. No one believed in us from the beginning of the season," he told a post-match news conference.
"We are so close to getting into the top four and on June 1 we will play Liverpool in the final in Madrid.
"It is an amazing season, in our new home, and I think that this chapter we are going to extend until the June 1, it will be amazing to close the chapter playing the final.
"If we win it will be an amazing book."
Lucas has often been a bit-part player since joining Spurs from Paris Saint-Germain last season but Harry Kane's latest ankle injury has thrust him into a prominent role.
Pochettino felt the star turn produced by the unsung Lucas spoke volumes for the collective spirit he has worked hard to nurture over the course of five seasons in north London.
"If our players are superheroes, he was a super, super, superhero. He was amazing," he said.
"He was really clinical in front of goal. It was a magic night for him. It is so important that after five years we know each other.
"To achieve big things it is very clear that you need 25 players and you need to show respect.
"That belief in all the squad paid off. When Lucas Moura arrived, it was a difficult and tough moment, always we believed to push them to be better.
"Of course, it was a tough period for him but we always respected him and were honest, but look at today without Harry Kane… many people were talking about Sonny [Son Heung-min] but it was Lucas who scored three goals.
"Maybe with 11 players you can win some games but to be in the final of the Champions League you need 24 or 25. We were always honest with everyone because some games you will need to use them and need them to give their best."
Pochettino abruptly ended his briefing with a "ciao" when asked whether Kane would be fit to face Liverpool.
Before that, he paid a warm tribute to Klopp's own shock troops.
"I want to congratulate Jurgen Klopp and all their players because I think they were amazing," he added.
"I think they are heroes too and it is going to be an amazing final we are going to enjoy a lot."