It continues to look like a case of when rather than if the Three Lions seal Euro 2020 qualification after their third Group A match ended in a third high-scoring win.
Five-goal hauls in March’s fixtures against the Czech Republic and Montenegro were followed by a 4-0 win against Bulgaria at a sold-out Wembley.
Kane was named man of the match after assisting Raheem Sterling and scoring the three others, with his first-half strike followed by a pair of penalties as he took his England tally to 25 goals in 40 appearances.
Kane’s treble took him ahead of Stan Mortensen and World Cup winner Sir Geoff Hurst in England’s all-time scoring chart, while Wayne Rooney’s record of 53 goals looks eminently achievable.
“I can only talk glowingly about that mentality to want to be a top player,” Southgate said. “Selfish isn’t the right word because he’s actually a very unselfish player.
“He works so hard for the team and he’s more than happy to drop into areas to leave space for others and play passes into others.
“But, when he gets his moment, then he just has an outstanding mindset and, technically, he’s a top, top finisher.
“But I go back to the fact that’s hours and hours of practice and if you talk to some of the other forwards in the squad, they would talk to you about how big an impression that has had on them and the way they practice when they go back to their clubs.
“Sometimes you can take part in finishing practices that can be a little bit half-hearted or not as focused as they might be - but he’s on it every single time.
“That’s where you get the transfer into the game.”
Manager Southgate lauded Kane’s “supreme temperament and technique” and also praised Sterling.
“I thought that he was outstanding, in the first half in particular where it was difficult to find space,” he said. “We were a little slow moving the ball.
“I didn’t think we had enough runs in behind but he was a constant threat throughout that period and his willingness to work without the ball to go and press and win the ball back for the first goal set absolutely the right tone for the team.
“So he’ll probably have the hump that he’s only got one today because his mindset has shifted so much towards that desire to score goals.
“Two years ago when we were working with him he was still a dangerous player but he just didn’t have that same desire to get into those areas and that ruthlessness in front of goal or that mindset of disappointment when he doesn’t score.
“And I am delighted for him because he has for such a long time been such an important part of our team and what we do.
“Now he’s reaping the personal gain of the goals and also the acknowledgement publicly that he should get.”
Kane and Sterling could get the chance to grab more goals in Tuesday’s Euro 2020 qualifier against Kosovo, even if it is likely to be a tougher challenge.
“I think Tuesday’s game will be completely different by the way,” Southgate said.
“I think that they are a really good side, so we have not had the tight, tense matches that the Nations League provided as of yet, that really we learnt so much more from.
“So, therefore, we’ve got to do that in training and the challenge of training has got to be so high that we learn from those moments and we can see what the players are capable of. “
Meanwhile, Bulgaria counterpart Krasimir Balakov lauded Kane as “one of the best in the world” and had no argument with the result, even if individual errors left him frustrated.
“Before the game at the press conference (on Friday), I said that there is an enormous difference in the levels of the two teams,” the head coach said.
“We were aware that England are an extremely good side and that’s why we wanted to be as compact as possible.
“But I think all the four goals were our individual results, even though I think the result was fair.”