Pochettino praised Alli, whose discipline came under the spotlight again for a strong early challenge on Charlie Taylor that had the Burnley fans baying for a red card.
Michael Oliver opted for yellow but Alli was booed loudly throughout following the penalty incident.
Pochettino said: “His personality and character, he doesn’t care what happens around him. He plays in the same way. That is why he has achieved all that he’s achieved in a very short period.
“I think it was yellow. I am happy with his performance and contribution. The penalty was a penalty and then he was always into the game trying to help the team and it’s coming.”
Burnley boss Sean Dyche felt both decisions were questionable but had no argument with the result and was awed by Kane’s performance.
“There’s some frustration about decisions today, but not decisions that were necessarily going to change the event, only a possible sending off because they were excellent today, simple as that,” said Dyche.
“Some of the referees talk about that orange card and it’s probably one of them. Kevin doesn’t need to challenge, but he (Alli) gets a toe on it and he’s going down and then the contact comes.