"There aren't many players who have scored a hat-trick at the World Cup, so that's why I'm proud", said Kane.
The in-form Tottenham striker bagged his third after a 61st-minute shot by Ruben Loftus-Cheek hit his foot and bounced over despairing Panama goalkeeper Jaime Penedo.
Intriguingly, it also means they are level on points, goal difference and goals scored with Belgium in Group G. The two sides meet in Kaliningrad on Thursday.
Cristiano Ronaldo and Romelu Lukaku both have four goals apiece and can be backed at [4.4] and [6.4] respectively, while Spain's Diego Costa (three goals) is available to back at [7.8].
It was the second hat-trick of the World Cup with Ronaldo having put the ball past David De Gea three times in Portugal's opening fixture. We wouldn't swap him for anyone in terms of number nines.
On the prospect of winning the Golden Boot, Kane said: "There's still a long way to go".
"It could have been a tough game but we started well".
"The referee was on top on things and we're proud of the way we played".
"It's unbelievable, I'm extremely proud".
England went into half-time with a 5-0 lead and each team scored once in the second half, leading Southgate to say it was an odd game after the high emotions of Kane's late victor to beat Tunisia in their first game.
He even managed to douse the flames of the first media storm of England's World Cup on the eve of the Panama match - over whether journalists should reveal his team selections by recognising the press should not be cheerleaders.
"But I would have loved a clean sheet, I thought we defended so well and dominated the match and to not get a clean sheet is a bit disappointing".
But Southgate said: "We would want to keep momentum and keep progressing as a team".
# This is only the fifth time in World Cup history that a team has scored 5+ goals in the first half of a match, with Germany vs Brazil in the 2014 semi-final the most recent occasion. "You know that when he gets opportunities he's going to bury them".