John Stones also scored twice while Jesse Lingard sent in a fine curler, and Kane gave credit to his teammates rather than celebrating the first World Cup hat trick for an England player since Gary Lineker did it in 1986.
The Tottenham forward scored twice from the spot before a Ruben Loftus-Cheek shot deflected off the striker's ankle in the second period.
It is predicted a wave of fans will dash to Russian Federation ahead of England's last group game with Belgium on Thursday in Kalinigrad. "We wouldn't swap him for anyone in the tournament in terms of number nines".
On the prospect of winning the Golden Boot, Kane said: "There's still a long way to go. It's not every day we get to go through after two games".
So if you had two incredibly cynical managers in charge of England and Belgium, both teams could play for a goalless draw and try to get their entire team, plus substitutes if required, booked, in an attempt to swerve the favourites.
"Extremely proud of the way we played, the hard work. and we also had fun out there as well which was important".
Their previous best was four goals against West Germany in World Cup final 1996.
Russia 2018: Russia, Uruguay advance to Round of Sixteen
Saudi Arabia still smarting from their opening day 5-0 drubbing by Russian Federation could make wholesale changes as they desperately try to stay in the tournament.
Brazil vs. Costa Rica
As the minutes ticked toward the end of regulation, Costa Rica , disciplined and organised in a five-man back line, held firm. Corinthians right-back Fagner replaced the injured Danilo in Brazil's only change.
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"Very unusual because I think we probably at times played better the other night than we have done today, but today we were really ruthless in front of goal".
"Just before half-time the game was done today, so it was a very, very odd feeling watching the second half, just trying to encourage the players to keep professional in the way that they played and be a top team by being ruthless".
The Group A winners will play the Group B runner-up, after which Group B winners will play Group A's second place finisher.
For England fans, the pain of being knocked out in the last 16 by Iceland in the 2016 European Championship and a first-round exit without a win at the 2014 World Cup, was finally slipping away.
"It's unusual because I enjoyed the win against Tunisia more - because of the tension and the fact you get over the line", said Southgate.
"We're a team who want to play and you see that on the pitch from the back to the front". As things stand they top the group having had less yellow cards than Roberto Martinez's men this tournament, having scored and conceded the same amount of goals.