West Ham may have been unable to match Tottenham's progress in recent years, but they have had the edge over their London rivals on home turf.
Twice in the last two seasons, a 1-0 home win for West Ham has brought an end to Spurs' title challenge.
Kane scored his first in the 35th minute.
Carroll had only been on the pitch for a couple of minutes, replacing the injured Michail Antonio, when his misplaced pass inadvertently helped set up the opener.
Kane's goals arrived inside four minutes of each other, with Christian Eriksen rewarded for a neat performance with a well-taken effort that makes him the Premier League's top-scoring Dane with 33 strikes. The rebound granted Kane an open goal, and he stroked calmly home.
It certainly had the desired effect on Kane, who had virtually polished off West Ham before half-time in the lunchtime kick-off.
When Christian Eriksen added a third - after Kane had thumped a free-kick against a post - Tottenham were cruising.
Kane again hit the woodwork soon after, but the Hammers suddenly had life when Hernandez was allowed to roam the six-yard box unchecked and he headed home Jose Fonte's flick.
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The Ivory Coast worldwide was dismissed on his first Premier League start for a needless second bookable offence with his team leading 3-1.
Tottenham manager Pochettino had taken a 49-strong group of players and club staff out for a mid-week meal at a London steak restaurant to strengthen the bond within the squad.
"We know we have to improve our home form and we will be right up there".
"Then we lose one ball, with enough players behind, and it was a brilliant finish".
The warning signs, though, were there for West Ham in the 24th minute when Aurier was sent clear on the right by Moussa Sissoko.
It was all too easy for Pochettino's team, who continued to apply intense pressure after the break.
'I think the first yellow was not a yellow but of course disappointed because he got the second and a red but that's football. And he was little use as a target man until that flurry.
By the time Hernandez's goal came, Carroll had been banished to the left flank.
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