Spurs cut Chelsea's lead at the top of the table to seven points with a hard fought win over Ronald Koeman's side at White Hart Lane.
And it was Kane who broke the Toffees' resistance with a long-range effort before tucking home a second goal just before the hour mark.
The 23-year-old also had a glorious opportunity to complete his hat-trick late on but saw his chipped shot saved well by Joel Robles.
Kane's exploits were enough to help him climb to the top of the Premier League goalscoring charts but Pochettino says the England international was disappointed about missing out on what would have been a third treble in four games.
"He had the possibility - it was a big chance but the keeper saved and he was a little bit upset.
"When you have the chance you have to kill the opponent always. He tried but missed.
"That's a good mentality - when you score twice and win but are always wanting more," Pochettino added.
Kane's form in front of goal is perhaps even more impressive given he was forced to miss a chunk of the campaign due to ankle ligament problem.
It has been suggested a brief reprieve from competitive action has benefited Kane but Pochettino is adamant the striker's rich vein of form is simply down to hard work.
"He feels fresh because he's trained a lot. He's had a lot of double sessions and takes care of himself.
"To be fresh you need to work hard every day and if you sleep and eat and forget to train maybe you are like me - fat!
"It's about consistency. He's in a very good level but not only now.
"If you look back and follow him when he started to play regularly in 2014 his stats are unbelievable. He is showing he is one of the best strikers," Pochettino insisted.