Jimmy Greaves’ Tottenham record had stood since 1970, but Kane’s first goal in his last three appearances gave him a share of the historic milestone.
Kane played despite being ill with a fever and Tottenham boss Conte admitted he is desperate to reward the selfless star with silverware.
Without a major trophy for club and country, Kane has been linked with Manchester United.
But Conte said: “Today he was amazing. I think he scored a fantastic goal, the execution, to control the ball, then to kick it in that way, only a world-class striker can score this goal.
“About the spirit that I spoke before, I want to underline Harry played with a fever and he was not so good, but he wanted to play because he understood the moment.
“He understood he is a point of reference for us. For sure, I would like to help him and me, also his team-mates, to try to do something important, to win something with Tottenham because he loves Tottenham.
“Tottenham is in his heart and it should be good if together we were able to win something, because then it is important to have this record, but I think it could be more important if you win a trophy.”
Kane could break Greaves’ record in Tottenham’s next match in the FA Cup fourth round at second tier Preston on Saturday.
It is only a matter of time before Kane takes sole possession of Tottenham’s most hallowed record.
That remarkable achievement would have seemed impossible during a difficult start to his career that featured several failed loan spells before he finally emerged as a star.
Greaves, who died aged 81 in 2021, is regarded as one of the most natural goal scorers in football history.
A member of England’s 1966 World Cup-winning squad, who netted 44 times for his country, Greaves is the highest scorer in English top-flight history with 357 goals.
For Kane to be alongside such a legendary player is a tribute to his predatory instincts and his perseverance.
The 29-year-old is now one short of becoming the third player to net 200 Premier League goals, after Alan Shearer (260) and Wayne Rooney (208).
Kane’s 18th goal
Kane also drew level with Rooney as England’s joint record goalscorer after bagging his 53rd international goal in the World Cup quarter-final defeat against France in December.
Thanks to Kane’s 18th goal in all competitions this season, fifth placed Tottenham closed the gap on fourth placed Manchester United to three points.
It was a much-needed result after successive defeats against Manchester City and Arsenal damaged Tottenham’s bid to qualify for next season’s Champions League via a top four finish.
Conte’s team had conceded 21 goals in their previous 10 league matches.
So, while Kane rightly took the spotlight, Conte was just as pleased by his players’ defensive resilience.
“It was a big win. I had a good answer from my players. I asked them for solidity of the last season and the will and desire to fight and to be resilient,” Conte said.
Failing to turn their first half pressure into goals proved fatal for Fulham as Kane entered the record books in first half stoppage-time.
Turning on the edge of the Fulham area, Kane worked a yard of space before curling a superb strike past Bernd Leno.
The record-equalling goal was met with sarcastic chants of ‘one-season wonder’ from Tottenham fans, who recalled the doubting response to his first prolific season for the club.
Kane almost had the outright record in the second half, but Leno tipped over his close-range header.
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