The 20-year-old opened his account from the penalty spot on Thursday night, before twice finding the net from one-on-one situations.
"Before the game I said that I wanted to score goals and to come away with a hat-trick is very pleasing," Kane told Spurs' official website.
"If you score a hat-trick at any level it's something special but to do it for England Under-21s, one step before the seniors, it's even better. Hopefully I can push on from here and score plenty more goals."
Kane has now scored for England at Under-17, Under-19, Under-20 and Under-21 levels.