Kane's well-taken penalty in the 2-1 win over Italy in the qualifiers for EURO 2024 saw him overtake Wayne Rooney as his nation's all-time top scorer on 54 goals.
Declan Rice had earlier given England the lead and debutant Mateo Retegui halved the deficit for the hosts at the Stadio Diego Armando Maradona on Thursday.
It is the first time England have beaten Italy away since 1961 as The Three Lions gained a measure of revenge for their EURO 2020 final defeat.
Kane told Channel 4: "It means everything. We're so excited to put the England shirt back on, get the campaign started for the Euros next year. It had to be a penalty, of course, but once it hit the back of the net there was so much emotion.
"It's a magical moment. Thanks to everyone who's helped me get to this stage. It's a great night. We haven't won in Italy for so long. To score, and win this game, is just special."
Kane added: "I was on the pitch when Wayne broke the record. I know what it meant to him and I was so proud of him. I remember when I presented him the boot for breaking the record, he said he would be giving it back to me one day. A special man."
Rooney was first to congratulate Kane, with a message posted on Twitter that read: "Congratulations to Harry Kane on becoming England’s all-time leading goalscorer. I knew it wouldn't take long but that was quick. Great man, unbelievable goalscorer and an England legend. Congrats Harry."