Minutes after Matt Ritchie's piledriver had drawn the hosts level, Kane's clinical header on the hour put the visitors back in front, taking him to another landmark.
Kane sealed the points in the last minute with another header to take his career tally to 201.
"Harry Kane knows how important he is, he is incredible," said Spurs boss Jose Mourinho.
"After such a difficult recovery [from injury] he is playing every minute of every game, this is the fourth game with two days in between and Harry gave everything."
Son Heung-Min had given Tottenham the lead in the 27th minute with his 11th league goal of the campaign.
Tottenham rose to seventh on 55 points with two games remaining, one point behind sixth-placed Wolverhampton Wanderers in the race for a Europa League spot.
Kane's 201 club goals have come in 350 appearances in all competitions with 185 of them for Tottenham.
It was also a milestone of sorts for Spurs manager Mourinho who claimed a first league win at Newcastle.
"201 in 350 league games -- that's not bad!" Kane told BT Sport. "It's nice to get to that milestone but the points are the most important thing.
"Hopefully, I'll get a couple of more before the season ends.
"A tough game and three massive points. We didn't play as well as we could have but we got the result in the end."
Newcastle were unfortunate to be down at halftime, although they were culpable for Son's strike.
Ritchie scored a deserved equaliser in the 56th minute when he gathered a poor clearance from Serge Aurier before lashing a fizzing shot past Hugo Lloris. Aurier was playing despite his brother being murdered in France this week.
Newcastle were only level for three minutes, however, with Spurs substitute Steven Bergwijn's first contribution being a superb cross that was met by Kane with an angled header.
Kane dived in to head his second after substitute Erik Lamela's shot was parried.