England captain Kane netted on the stroke of half-time with a clinical strike from the edge of the penalty area at Craven Cottage.
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.
"I was just trying to get myself into a position where I can get half a yard. I used the defender to bend it around him. It was a nice goal," Kane said.
"I have been playing football long enough now to know where the goal is. It is just instinct."
Kane could break Greaves' record in Tottenham's next match in the FA Cup fourth round at second tier Preston on Saturday.
"The FA Cup is important for us. We will see what the manager says but I will be ready," Kane said.
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 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.
Antonio Conte's team had conceded 21 goals in their previous 10 league matches, losing six of those, and causing the Italian to claim they were defending like a "relegation team".
So, while Kane rightly took the spotlight, Conte will have been just as pleased by his players' defensive resilience.
Kick-off was delayed by 15 minutes due to "local travel restrictions" on the underground service in west London.
But Kane ensured a freezing evening by the River Thames would be a memorable one -- after Tottenham survived a nervy start.
Hugo Lloris has made a number of costly mistakes this season, but Tottenham's keeper was equal to Andreas Pereira's free-kick.
Fulham should have taken the lead when Pereira whipped a cross towards Harrison Reed, whose volley was straight at Lloris from eight yards.
With Tottenham struggling to stem the Fulham tide of attacks, Aleksandar Mitrovic's header from Kenny Tete's cross flashed just wide.
Fulham winger Willian went close to breaking the deadlock just before half-time with a stinging strike that Lloris pushed away.
Failing to turn the 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.