Kane netted a hat-trick, as the Three Lions hammered Panama 6-1 on Sunday afternoon, to book their place in the final 16.
"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 Tottenham striker bagged his third after a 61st minute shot by Ruben Loftus-Cheek hit his foot and bounced over Panama goalkeeper Jaime Penedo.
The scoreline meant England are joint on points and goal difference with Belgium, but ahead on their fair-play record.
Kane now tops the scoring charts at the World Cup, one ahead of Manchester United's Romelu Lukaku and Real Madrid's Ronaldo following the latter's hattrick against Spain and single strike against Morocco.
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.
"It's going well, there's still a long way to go, the most important thing is winning games", Kane said.
"It could have been a tough game but we started well".
Ronaldo scores fourth goal, gives Portugal early lead
Amrabat was furious at not being awarded a penalty after wrestling with Raphael Guerreiro and the entire squad erupted in outrage. Despite their battling display, Morocco will rue the unforgivable defending that made all their subsequent good work irrelevant.
England manager Gareth Southgate has heaped praise on the Russian Federation 2018 World Cup infrastructure and stadiums and admits he and his team have "been really looked after" in the country. "We had fun out there too", said Kane, calling his third "one of the luckiest goals of my life". "It's not every time you go through after two games, so we're happy with that".
"There's a lot of good players behind me, hopefully I can take it into Thursday's game".
"We would want to keep momentum and keep progressing as a team", Southgate added.
They will then go on to the quarter-finals, where they will play either the runner-up or victor of Group H, depending on where they finish.
The four-times world champs suffered a shock 1-0 defeat to Mexico in their first group game, and after the South Americans won for a second time against South Korea earlier in the day Germany knew that only a victory over Sweden would realistically keep them in this year's tournament.
"It's unusual because I enjoyed the win against Tunisia more - because of the tension and the fact you get over the line", said Southgate.
"And John Stones is such a threat from set-pieces so we're delighted he's got two goals and he'll be buzzing about that".