The London side were down 3-0 on aggregate midway through their semi-final second-leg against Ajax in Amsterdam, but a Lucas Moura hat-trick in the second-half booked their spot in next month's final against Liverpool.
Ajax looked capable of grinding out the result until the final seconds of the game, as Moura struck the decisive six minutes into stoppage time.
The final whistle sent the Tottenham players and staff into wild celebrations on the field and coach Pochettino was in tears as he embraced his players.Speaking after the game, he was still overwhelmed by his side's achievement and continued to praise his players.
"It is still difficult to call. It's emotional. It's amazing. Thank you football," he told BT Sport. "Thank you these guys, my players are heroes. In the last year I have been telling everyone that I have a group of players who are heroes.
"The second half was amazing. This type of emotion, without football, it's impossible to live. I am so emotional now. Thank you to our fans and the people who believed in us from the beginning of the season. I think it's fantastic. It's difficult to describe.
"We were talking before the game that when you work and when you feel the love, it's not stress, it's a passion. I think the emotion we showed today was passion. We love this sport that is football.
"I think the game against Man City [in the quarter-finals] and today, it's amazing. For us it was so tough, it was difficult to compete at this level. It's a massive competition. I am so grateful to be a coach, to be in football, and to be leading this type of team."
While the Argentine was proud of all of his players, he said his side's hat-trick hero was a cut above the rest.
"All heroes, but [Moura] is a superhero. Unbelievable hat-trick. He deserved it. He's a good guy," he added. "I think everyone deserved it from the first one to the last one in this club. To be in this moment right here, to have the chance to play in the final is amazing.
"I want to remember my family, it's for them too. It's for the people who passed, in that moment, it's amazing to reward them. We need to be ready for the next game on Sunday [against Everton] and be ready then for the final in Madrid."