Kane netted his 54th goal for England in his 81st international appearance from the penalty spot in Naples after Declan Rice had fired the visitors into an early lead.
The England captain moves clear of the record mark he had shared with Wayne Rooney.
“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 so to score, and win this game, is just special,” said Kane.
Gareth Southgate’s men gained some measure of revenge for their defeat by the Azzurri in the Euro 2020 final.
In their first match since bowing out in agonising fashion to France in the World Cup quarter-finals, England were dominant in the first 45 minutes.
Rice rifled home his third international goal on 13 minutes after Italy failed to clear a corner.
Another set-piece led to Kane’s record-breaking moment as Giovanni Di Lorenzo handled inside the area.
The Tottenham striker banished the memories of his missed penalty late on against France in Doha by sending Gianluigi Donnarumma the wrong way.
But Italy rallied after the break when Argentine-born striker Mateo Retegui fired home on his international debut.
England were forced to see out the final 10 minutes a man down after two quickfire yellow cards for Luke Shaw, but they held out for their first win away to Italy since 1961.
“We feel like we are one of the best teams in Europe,” added Kane. “Coming here and putting in a performance like that today shows we are ready for the next challenge.”
In the other game in Group C, North Macedonia beat Malta 2-1.
Ronaldo marks historic night
Ronaldo marked Roberto Martinez’s bow as Portugal boss with another historic night in Lisbon.
The 38-year-old, who was dropped from the Portuguese starting XI during the World Cup, set a new record for men’s international appearances by passing the previous mark of 196 caps held by Kuwait’s Bader Al-Mutawa.
Joao Cancelo fired home the opening goal after just eight minutes but it was not until Bernardo Silva doubled Portugal’s lead early in the second half that the floodgates opened.
Ronaldo made it 3-0 from the penalty spot before blasting home a free-kick from the edge of the area for his 120th international goal.
“It’s a new cycle. It is very important that a player shows commitment and that he can use his experience,” said Martinez. “Cristiano has that: a lot of experience and a lot of commitment to the national team.”
In the same group, Rade Krunic scored twice as Bosnia and Herzegovina swept aside Iceland 3-0.
But Slovakia, who have qualified for the last two Euros, were held 0-0 at home by Luxembourg.
Euro 2020 semi-finalists Denmark bounced back from their disappointing early exit at the World Cup to beat Finland 3-1 in Copenhagen.
Atalanta forward Rasmus Hojlund scored a hat-trick for the Danes on his first start for his country.
Elsewhere in Group H, Dion Charles scored twice as Northern Ireland won 2-0 in San Marino, while Slovenia came from behind to win 2-1 away to Kazakhstan.
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