Spurs looked set to claim back-to-back Premier League victories for the first time since November thanks to Kane's 25th-minute effort – his 36th league goal against a London rival.
Scott Parker was enraged by the decision to reschedule his team's trip to Tottenham Hotspur Stadium at such short notice, but Fulham stayed in the game and, with Son Heung-min squandering a great chance to wrap things up, got its reward.
Substitute Ademola Lookman proved the difference-maker as he pinged in a cross that was met by the towering Cavaleiro to seal a share of the spoils.
Sergio Reguilon blazed over inside the opening 10 minutes, but Fulham showed plenty of intent and Hugo Lloris had his palms stung by Andre-Frank Zambo Anguissa's strike.
Alphonse Areola came to the fore with two superb stops, both from Son, who prodded goal-wards from point-blank range before attempting to guide a header into the top-right corner.
Areola was beaten in the 25th minute, though. Where Son went for height, Kane went low, diving to meet Reguilon's exquisite left-wing cross and head in off the post.
Kane was the beneficiary of another terrific cross – this time from Serge Aurier – soon after, but failed to keep his header down.
Tosin Adarabioyo made a fantastic last-ditch tackle to frustrate Moussa Sissoko after the restart, while Areola had to lunge across to keep out Harry Winks' dipping effort.
Son could have wrapped things up with just under 20 minutes remaining, yet hit the base of the right-hand upright, and Spurs were made to pay.
Cavaleiro towered over Eric Dier to meet Lookman's cross and send a powerful header beyond Lloris, rescuing a hard-earned point with Fulham's first shot on target of the half, with Reguilon rightly denied a last-gasp winner by the offside flag.