Victory left Spurs level on points with fourth-placed Liverpool, who face runaway Premier League leaders Manchester City on Sunday, in a competitive race for a place in next season's Champions League.
He took the ball around Jonjoe Kenny before bursting down the left-hand side, with his cross into the box tapped home by a waiting Kane as he equalled Sheringham's record.
It was a remarkable December for Kane as he scored eight goals in the league in six matches.
That was his 97th Premier League goal - and his 98th came only 12 minutes later as he scuffed a shot into the net from Dier's cross.
The England worldwide now has 20 goals this term to leave him just 11 shy of the leading number of strikes in a 38-game Premier League season - a record held by Alan Shearer, Cristiano Ronaldo and Luis Suarez - and 14 away from Andy Cole and Shearer's joint haul of 34 during a 42-match campaign.
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.
"He can achieve all that he wants because of his willingness to work".
"He is always thinking "improve, improve, improve and how to score goals and to improve the team" because his mentality is always collective and that is so important to explain to people".
'Obviously there's a lot of talk before the game about whether I'd do it or not so to get that goal early on in the game as well, it was a great achievement'.
"Maybe I have a bit of responsibility by playing too much attacking-minded players and not as many defending-minded players".
Chelsea was unable to pierce Leicester's defence even after visiting left-back Ben Chilwell was sent off for a second bookable offense midway through the second half.
Tottenham Hotspur's Harry Kane celebrates scoring again for his side (pic: Anthony Devlin/PA Images). The Baggies secured their first top-flight win in 21 games with a 2-0 result against Brighton and Hove Albion on Saturday.
He did have a rare off-day against West Ham in Spurs' last Premier League outing though, with the Hammers digging deep to get a 1-1 draw.