Kane broke the deadlock three minutes into the second half Crystal Palace, then took just five more to bag his brace for his 264th in a Spurs shirt and leave him just two shy of tying Greaves' club record of 266.
Kane's 15th Premier League goal of the season - six fewer than Manchester City's Erling Haaland - moves him to 198 in the competition's history, with only Alan Shearer (260) and Wayne Rooney (208) ahead of him.
And Antonio Conte, who was relieved his side picked up their first victory since the World Cup break, suggested more milestones are simply inevitable for the England captain.
The Tottenham boss said: "I think to speak about Harry, I think a bit that we waste time to explain about the type of player that he is. I repeat, we are talking about a really world-class striker and he's going to beat every record with the goals.
"For us he's an important point of reference, not only for the goal but also on the pitch and with Hugo (Lloris) our captain. Especially in the difficult moments, Harry, you know that he is ready to fight, ready to help the team-mates, ready to solve the situation with a goal, with an assist.
"To continue this way, to face every situation positive or negative, always with a great spirit, great character, to overcome the difficulty I think in this period and also before the start of the World Cup."
Struggling Spurs were hoping to bounce back from a 2-0 defeat to Aston Villa on New Year's Day.
They were thwarted in the first period by the resilient hosts, who often looked the brighter side but struggled to find the finishing touch.
It was goalless when the sides returned for the second half, but Kane quickly shifted momentum when he nodded in Ivan Perisic's cross, then sent a low strike past Vicente Guaita to double his side's advantage.
Son Heung-min set up Matt Doherty for Spurs' third before getting his name on the scoresheet in the 72nd minute.
Victory ensured Tottenham closed the gap to fourth-placed Manchester United to just two points.
Conte added: "To recover three days after a defeat at home where the expectation was to try to win, and to go again in a really good position in the table was not easy for the atmosphere, for the players, but the response today was really good because I think my players showed me every time great character, great personality in every situation."
Palace boss Patrick Vieira was pleased with his side's first-half performance, and refused to liken the loss to his side's 3-0 defeat at home to Fulham on Boxing Day.
"I will not put those two games together," he insisted. "I think the Fulham game, everything went wrong, we had two sent off and our attitude wasn't right.
"Today I think the way we played the game, I think we had the right approach, we had the right attitude, we were toe-to-toe against them. We created maybe the best chance in the first half and we had a really good period. If we scored the goal before half-time, maybe the game would be different."