The result sees Antonio Conte's side remain in fourth place in the table and stretch their gap over rivals Arsenal to three points.
It turned out to be a great day for Spurs with top-four contenders Manchester United and Arsenal also losing on Saturday.
"I'm so happy to get three points, it was a hard afternoon," Son told BBC in his postmatch interview. "The first goal was important, early goal to change the game. Without Hugo we were probably 3-1 down in the first half, Hugo saved us today and it was a big performance from our captain when we needed him and he turned up saving every ball. He was unbelievable.
"We knew about the other results of Arsenal and Manchester United -- when we did the warm-up the fans were celebrating so we knew but we don't want to focus on that result. They are playing their game and we're playing ours so we don't want to focus on other teams because it's different. We're in different situations, we have to focus our performances and be calm, cool and we did that today."
Spurs took the lead inside the opening three minutes when Son reacted quickest to a blocked shot and smashed the ball into the bottom corner.
Villa were the better team following the goal and the hosts missed a number of chances to equalise.
Spurs doubled their lead five minutes after the break when Son set up Kulusevski who produced a fine finish into the corner.
Son completed his hat trick five minutes later as Kulusevski returned the favour with the South Korean's shot going in off the post.
"Hugo was amazing -- we are talking about a keeper that has won a World Cup and everything with France, the captain of his national team and us. Important for us because of his experience, today we needed him and he answered very well," Conte told BBC after the game.
"We created a good chemistry between ourselves and we can speak about Son, but don't forget Harry and we have players on the bench who are important in Lucas [Moura] and Steven Bergwijn. We need to continue to work with this application and intensity and desire because the club deserve much better than fighting for fifth, sixth or seventh. I like to be competitive."