Raheem Sterling thought he had completed his hat-trick in added time to send City through 5-4 on aggregate, but the goal was dramatically ruled out for offside.
Christian Eriksen gave the ball away in his own half but Bernardo Silva touched the ball on to Sergio Aguero, meaning the forward was offside before squaring to Sterling.
Sissoko, who was replaced in the first half due to injury, revealed he was unaware Sterling's late strike had been disallowed as he had gone to the dressing room.
"At 4-3, it's extra-time, we thought we were going to do it, then I'm on the bench, and they scored this goal for 5-3," Sissoko told reporters.
"When I see that they score, I'm in all my mood, I go back to the dressing room so I didn't see that the goal had been cancelled, so in my head we were eliminated.
"And it was one of the staff members who told me, when he came back to the dressing room, "Incredible! We did it!'', I said, 'How did we do it?', he said, 'Yes, the goal was cancelled!' So at that moment I put on a T-shirt, I ran outside to celebrate with my team-mates.
"I forgot the injury and ran away because it was an emotional shock. It's a great thing, to be qualified for the semi-final, for us players and the club, it's something historic. I had to celebrate this qualification with everyone."
Tottenham lost their first two group stage matches but bounced back brilliantly to reach the knockout rounds, with Ajax now awaiting in the semi-finals after they downed City on away goals after Wednesday's 4-3 second-leg loss.
Captain Hugo Lloris, though, says Pochettino's side can go even further having reached the semi-finals of the Champions League for the first time in the club's history.
"Anything is possible at this stage of the competition, qualifying as we did... beat Borussia Dortmund, Manchester City," Lloris said.
"Even if they [Ajax] are still young players, they are already a great team. We have to take all the chances on our side to believe in the achievement.
"For the club, it's historic but we don't want to stop here."
City boss Pep Guardiola said it was a "cruel" way to settle a thrilling Champions League quarter-final, ending their quadruple hopes in the process, but Bernardo Silva defended the use of VAR.
Before Sterling's late goal was ruled out for offside, VAR checked a Spurs goal by Fernando Llorente that seemed to brush his arm before bouncing in off his leg.
"I think VAR is a good thing because it helps the referees make better decisions, so no problems with the VAR," Silva told reporters.
"Obviously I've had the opportunity to watch the last goal from Tottenham and the ball hits the hand [of Llorente]. No doubt about that, then the referee took his decision, so we have to respect it."