Spurs trailed 3-0 on aggregate at the interval of Wednesday's semi-final second leg, but Lucas Moura scored three goals in the second half - competing his hat-trick in the sixth minute of added time - to send his side through to the final on away goals.
Speaking after the match, right-back Kieran Trippier revealed vice-captain Kane - not part of Tottenham's matchday squad because of an ankle injury - went "mental" in the dressing room.
But Pochettino says Trippier's comments may have been misunderstood and that skipper Hugo Lloris instead deserved praise for his role in inspiring the dramatic fightback.
"It surprised me," Pochettino said at a news conference.
"Listen to Trippier - he didn't say Harry was talking. It wasn't a team talk [from Harry]. He was there like Davinson Sanchez or another player not involved.
"Trippier said 'his presence'. Harry was there and translated confidence and of course emotion.
"But to be honest, I was talking. It wasn't a moment to talk too much. And in the end who lifted them before they went on the pitch was our captain, Hugo Lloris. I remember the words of Hugo Lloris as I was leaving the dressing room. He was shouting, 'We are close, we need to score one goal more, never give up and try because the moment we score we're going to be in the tie'."
Spurs went into half-time at the Johan Cruijff ArenA knowing that three second-half goals would be required to set up a meeting with Liverpool in the June 1 final.
Despite Tottenham having it all to do, though, Pochettino insists he was "relaxed" as his side did not deserve to be two goals behind on the day.
"We give five minutes to the masseur, physio, doctor and sports science. When I arrived, I talked and showed some clips," he said. "I think it was the most relaxed half-time in the last six months. I explained to Danny Rose today. Danny asked me today, 'Gaffer did you really believe at half-time we could achieve it?' I said, 'Yes, I was so relaxed'.
"He said, 'Were you surprised?' I said, 'Yes'. I was so upset with destiny at this moment. Because the game was even. The difference was they were clinical in front of goal, we conceded and we didn't score. We check everything - the possession, the metres we run, the attempts, the intensity.
"In five minutes we checked everything and then we need to explain to the players what we need to do in the second half. It was of course [Victor] Wanyama out, and Fernando [Llorente] in.
"We didn't deserve to be 2-0 down. That's why I said, 'Keep going, be relaxed'. It wasn't a team talk like Arsenal, when I nearly broke my hand and I needed to scream, and I needed to spark the players. Because the players were focused. It was a bit unfair.
"Harry was there, listening to us. Maybe Trippier loved Harry was there and for himself it translated a confidence. I promise our captain was very good, with the last word and before the game too, with the last word. It's always our captain who talks after me. Hugo was brilliant."
Tottenham face Everton in their final Premier League game of the season on Sunday, before switching focus to their Champions League final clash with Liverpool on June 1.