The visitors produced a scintillating display of attacking football to take full advantage of Leicester's 2-2 home draw with West Ham United on Sunday and close the gap to five points with four matches remaining.
Leicester remain favourites with 73 points but Tottenham's fourth victory in five games, achieved with a real flourish, means the title race is back on the boil.
"We are ready to put the pressure on Leicester. We think we can do it. Leicester are five points ahead but we have closed the gap, that's all we could do," Kane, who moved two goals clear of Leicester's Jamie Vardy at the top of the scoring charts with 24 goals, said.
"We scored four but could have had six or seven. It is one of our best performances all season. I was itching to play. I had to wait a whole day and a half."
England striker Kane gave Tottenham the perfect start after nine minutes, curling a textbook shot past keeper Shay Given after being played in by Moussa Dembele.
Stoke briefly threatened a response and Hugo Lloris made a fine save from Marko Arnautovic but Tottenham should have gone further ahead when Alli played in Christian Eriksen but the Dane curled a shot against the crossbar with the goal gaping.
More chances went begging early in the second half for Tottenham early in the second half but they doubled their lead after 67 minutes when Eriksen's pass sent Alli clear and he calmly chipped Given.
Incredibly Alli struck the post after rounding Given shortly afterwards, a comical miss that had Spurs manager Mauricio Pochettino beating the turf in frustration, but it did not prove costly as Kane calmly converted Erik Lamela's pull back after another rapid Spurs attack.
Tottenham were running riot and completed the rout after 82 minutes with Eriksen again the provider, clipping a clever pass to the outstanding Alli who swept home a volley to condemn Stoke to their joint worst home defeat in the Premier League.
(Reporting by Martyn Herman, editing by Pritha Sarkar)
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