Kane, the Premier League's top scorer with 18 goals and December's player of the month, recently suggested that Spurs have to start winning trophies if they want to retain his services amid reported interest by Real Madrid.
"It was a great performance, but it's not only today", Pochettino said of the forward.
For much of the match, however, Kane had played second fiddle to Son Heung-min, who broke the deadlock after 26 minutes and set up Kane's first shortly after the interval.
However, it did not take long for Tottenham to change that statistic and, on 47 minutes, Kane tapped home after smart work by Son on the left.
Harry Kane bagged a brace, becoming Spurs' all-time record Premier League goalscorer, while Heung-Min Son and Christian Eriksen were also on target as the North Londoners kept in touch with the top four with a scintillating display.
Everton had the ball in the net eight minutes later, but it was ruled out for offside.
Eriksen played the killer a pass, an absolutely brilliant ball out wide to Serge Aurier and he took one touch before delivered what appeared to be a shot.
There are not many thing Kane has failed to do for Spurs in terms of scoring - but one thing is score in five successive Premier League home games.
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He added the country "would be forced to take large numbers of people from high crime countries which are doing badly". The sources spoke on condition of anonymity because they weren't authorized to publicly describe the conversation.
He set a record of 39 Premier League goals in a calendar year in the process and has now joined the former Liverpool captain in the history books.
62' - Dominic Calvert-Lewin replaces Cenk Tosun for Everton.
"This is a historical tweet from years ago so he doesn't need any reminding because all the players have already been made aware for more than this year".
"It was a shocking second-half performance".
- This was Tottenham's biggest margin of victory in a game against Everton since a 10-4 win in October 1958.
Tottenham Hotspur's Harry Kane celebrates scoring again for his side (pic: Anthony Devlin/PA Images).
Allardyce suggested after the match that lessons had not been learnt from Everton's 2-1 FA Cup defeat by Liverpool last week, a game that was settled by Virgil van Dijk's late header.