The enforced departure of their talisman sparked Spurs into life, though, and they pulled level a minute later when Alli diverted Aurier's teasing cross past Begovic with his thigh for his first Premier League goal since January 2.
"I think it's a massive situation that you must learn [from]", Pochettino said, according to Sky Sports.
"Look, in football you need always some luck because the decision should be different, maybe now we are talking in a different way, but that is football, it happened Wednesday [against Juventus], happened the week before, but I accept football is like this".
Bournemouth started positively and nearly took the lead in the fourth minute when Junior Stanislas broke through on goal and rounded the keeper, but he saw his effort come back off the post. He'll undergo a scan tomorrow to see just how bad the injury is.
Pochettino was then asked about Dele Alli and Danny Rose's conditions as they were also substituted during the 4-1 victory, which was secured through a Son Heung-Min double and strikes from Alli and Serge Aurier.
Tottenham were down 1-0 at the time of Kane's withdrawal, though Dele Alli quickly equalised to make it 1-1 at the Vitality Stadium.
Tottenham's Son Heung-min celebrates after the match. "I believe those details can help the club".
"I thought we were a little bit naive the way we chased the game to try and get something from it".
Speaking about Alli, Pochettino added: "I don't believe that it's a big issue".
The result against Bournemouth means that Spurs are now third in the Premier League table with 61 points from 30 matches, just four points behind second-placed Manchester United.
And with Bournemouth pushing for an equaliser, the South Korean had acres of space to run through and slot home his 18th goal of the season three minutes from time. "A winning mentality is the most important".