With Spurs leading 1-0 from the first leg, goals from N'Golo Kante and Eden Hazard turned the tie around for Maurizio Sarri's Chelsea at Stamford Bridge.
Fernando Llorente headed home Danny Rose's cross to level the tie at 2-2 on aggregate, but Eric Dier and Lucas Moura missed from the spot to give Chelsea's a 4-2 penalty shoot-out triumph.
Previous Carabao Cup rules would have seen Spurs force extra-time and potentially progress on away goals with the tie levelled at 2-2, but the competition changed its setup this term.
And a proud Pochettino noted the change in the rules of the Carabao Cup ultimately cost injury-hit Spurs the chance to book a Wembley date with holders Manchester City.
"I think it's a competition. We came here to try and win the game and to try and be in the final," Pochettino told a news conference. "Congratulations to Chelsea. I think they deserved [to win].
"We made an unbelievable effort. This season we are fighting not only with the opponents with everything that's happened. That is not an excuse. It'll make us stronger. We have three more competitions, still. We arrived in a semi-final and lost to Chelsea, one of the best teams in Europe.
"I can only feel proud of the performance and the way we competed. When you arrive in this type of situation, you have to compete. We did that in both legs.
"In the Copa del Rey in Spain, and other competitions in Europe, if you score away from home... and it's 2-2 at the end of 180 minutes, I think the team who scored away from home goes through. Last season, they changed the rules and now we're out because of penalties.
"I was joking with [assistant] Jesus [Perez] about that. Only I feel proud. I told the players I feel proud. We have unbelievable players and squad, the performance of everyone from the team, and how we play... I can only congratulate them."
Dier scored a historic penalty to book England's place in the World Cup quarter-finals last year but he cleared the crossbar with his effort to give Chelsea the advantage.
David Luiz stroked home his kick to secure the Blues' place in the final and ensure Chelsea converted all four of their penalties, with Lucas having also been denied by Kepa Arrizabalaga, but Pochettino would not blame the Spurs duo for missing.
"Anyone can miss who takes the responsibility to shoot. For me, it's not a problem," he said. "It's about the emotions you have as you shoot. Sometimes you score, sometimes you miss.
"In the history of football, it's nothing to complain about. You can only feel pride for the players who take the ball and shoot. It's not a problem for me, the mistakes. I feel so proud. Only people who shoot can miss."
Pochettino also confirmed injuries that forced Moussa Sissoko and Ben Davies off were not serious, with Spurs already without striker Harry Kane, England midfielder Dele Alli and Son Heung-min, who is at the Asian Cup.