Spurs are once again in the position of chasing the Premier League leaders and are four points behind Chelsea after beating Bournemouth 4-0 on Saturday.
Mousa Dembele and Heung-min Son were also on target, while top-four hopefuls Manchester City gave their Champions League chances a lift with a 3-0 win at Southampton.
When asked if goal difference will be important this season, Pochettino said: "It's so important".
The goal makes Kane just the fourth player to score 20 Premier League goals in three consecutive seasons after Alan Shearer, Ruud van Nistelrooy and Thierry Henry.
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Mauricio Pochettino's Tottenham side are now four points behind Chelsea, who face Manchester United at Old Trafford on Sunday.
Tottenham have made a habit of killing off games early in recent weeks and when Son doubled his side's advantage after 18 minutes, the three points already looked secure. "It would be fantastic if tomorrow night, after the game, the gap is still four points". We are fighting for ourselves to be there. I want to do it year in, year out, and it's good to get three years in a row but we still have a lot to play for this season so I try not think about it too much.
Spurs went into the match on the back of 11 successive home wins and continued that impressive run in the blistering London sunshine. That levelled the record total for a South Korean player in a single European season, achieved by Cha Bum-Kun at German club Bayer Leverkusen in the 1985-86.
Naturally, though, on his first start in nearly five weeks, Kane netted his 25th goal in all competitions this season three minutes into the second half.
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Substitute Vincent Janssen rounded off the action with a fourth goal in stoppage time, the Dutchman's first Premier League strike from open play since joining previous year.
Goals by Robert Huth and Jamie Vardy put them 2-0 ahead but Palace hit back through Yohan Cabaye and Christian Benteke.
Swansea fell 1-0 to Watford to stay in the drop zone when Etienne Capoue scored the only goal of the game in the 42nd minute.
Fabio Borini pounced on Darren Randolph's error before Sam Byram was dismissed.
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Kane is in second place on the Premier League scoring table for the season to date, with Everton's Romelu Lukaku on 23 at the time of writing.