Kane doubled his side's lead after the break, slotting home after Mousa Dembele initially pinched the ball off Morgan Schneiderlin.
Ronald Koeman's Everton, on the other hand, stay seventh.
Everton had hope in the 81st minute though when Romelu Lukaku struck, though Dele Alli restored Spurs' two-goal advantage in injury time; Everton then claimed a consolation also in injury time through Enner Valencia.
Lukaku has notched his 61 goals from just 175 Premier League games, breaking the 60-goal record set by Duncan Ferguson.
Tottenham Hotspur and Manchester City both issued timely reminders to Chelsea that the Premier League title race might not be over with impressive victories on Sunday.
U.S. shale producers renew their challenge to OPEC: Kemp
To support this trend, Saudi Arabia has offered to reduce oil production even more if rival Iran caps its own output this year. However, some data other than the production numbers suggest that the Kingdom might be bypassing the OPEC deal by other means.
David Moyes' side gave as good as they got for most of the opening period with Jermain Defoe a constant threat. "For me Tottenham is one of the biggest clubs".
The Toffees have fewer injury problems, and have named a almost unchanged side for the third straight match, although manager Ronald Koeman has made one change with Gareth Barry coming in for Ademola Lookman.
Lukaku said he was trying not to think about his private battle with Kane going into Sunday's game. We are in a position where you can see we play very good football, exciting team, very good players.
Barkley featured for Everton in the Premier League match against Tottenham away from home at White Hart Lane in North London.
Jermain Defoe hit the post for Sunderland early on before Aguero tapped in City's opener three minutes before half-time.
Banksy opens dystopian West Bank hotel
The hotel features a presidential suite, a museum with the artist's work, a themed bar and interactive exhibits. Each of its nine rooms overlook the wall the Israeli government built within and around the occupied West Bank.
"If you compare the two teams of this afternoon its the third season of Pochettino at Tottenham".
On striker Harry Kane, Pochettino added: "He's at a very good level".
"We had the chance to get a point but it was not deserved".
"Possibly it's a mentality issue", Liverpool midfielder Adam Lallana said, "and something that we need to learn quickly because time will eventually run out".
Whilst Kane may not carry the same class that came with the likes of Klinsmann at Spurs, he makes up for it with his work rate and desire for the team which is just as important as his goals, particularly with the style of football Pochettino likes to play. It's easy to forget his tender age because he's been an established Premier League striker for four years now, but he still has plenty of time to improve and become one of the world's finest forwards.
EPA Won't Require Methane Data From Oil, Gas Companies
Some oil companies are already teaming up with tech developers to better manage the methane emissions at their fields. Environmentalists, meanwhile, are anxious that the oil and gas industry has Pruitt's EPA in its pocket.