Mauricio Pochettino's side were heading to their third defeat in the space of eight days, following reverses against Burnley and Chelsea, as Arsenal held firm after Aaron Ramsey's 16th-minute breakaway goal.
But Kane won and converted a 74th-minute penalty amid strong suspicions of offside – and that was only the start of the late drama.
Spurs defender Davinson Sanchez was penalised for a foul in the box on Pierre-Emerick Aubameyang, only for the former Borussia Dortmund star to see Hugo Lloris save his stoppage-time attempt from 12 yards.
There was still time for another Arsenal substitute, Lucas Torreira, to receive a red card for a foul on Danny Rose and Kane hailed Spurs' battling qualities as he reflected upon a breathless finish.
"We came out in the second half fighting and deserved to get back into the game," said Kane, who became the leading goalscorer in Premier League north London derbies.
"We put the pressure on – good aggression, good pressing, good tackling.
"Obviously they got the penalty at the end, thankfully Hugo saved it, and it was the least we deserved."
Referee Anthony Taylor found himself under the microscope as replays showed Sanchez's contact with Aubameyang to be minimal, but Lloris was reluctant to criticise a call that allowed him to be the hero.
"It's part of the game, it was a little bit crazy at the end," the France captain said.
"It could have been 2-1 for them or 2-1 for us and 1-1 is probably a good point for us, especially after the bad start we had in the first half and a tough week.
"I don't know [whether it was a penalty], everything went so quick. I will protect my defender always but I respect decision of the referee. Thankfully for us he didn't score."
Upon being told he was in an offside position from Christian Eriksen's free-kick before being barged over by Shkodran Mustafi, Kane suggested it was a break Tottenham deserved.
"It evens itself out throughout the season and throughout the game. We've had a few this year not gone our way and luckily we got one today," he added.
"We've had two disappointing results in the past week. It was a perfect game, the north London derby to show a bit of passion and show the fans we are ready to fight until the end of the season.
"It's a point. We wanted three but it stops the losing streak.