Lucas Moura's 85th-minute goal saw Spurs claim a 1-1 draw at Barcelona in Group B on Tuesday.
The manager went on to call the achievement the most "special" of his time in London, citing the four-and-a-half year journey to this point.
Many questioned Spurs' chances after they lost their opening two games and drew at PSV, and Pochettino hailed his side's achievement.
"I am so happy, so proud. I remember after Eindhoven nobody believed in us. They all said it was mission impossible, but we are here," he told a news conference.
"We are in the next stage and it is a massive achievement for the club. We are so happy for our fans that were here, happy for our fans in London and around the world.
"It's a massive boost for the club and to have the idea to play in the next stage next year in the new stadium is a massive motivation for us."
Ousmane Dembele's fine individual effort had given Barca the lead before Lucas' late equaliser.
Spurs are next in action on Saturday when they host Burnley at Wembley Stadium.