The match got off to a slow start until Tottenham found their opener just after 25 minutes when a ball into the box found Kane alone in front of goal and the England international made no mistake with his strike to be Everton keeper Jordan Pickford.
The lead wouldn't last long for Jose Mourinho's side, though, as Everton drew level from the penalty spot through Sigurdsson following a clumsy foul by Sergio Reguilon on James Rodriguez in the area with the first half ending 1-1.
There was a greater sense of urgency from both teams in the second half and it was Everton that took the lead after 60 minutes when Sigurdsson finished off a classy team move with an equally classy first-time finish to put Carlo Ancelotti's team up 2-1.
The visitors were undeterred and got back on level terms with Kane's second goal of the night, dispatching a clinical volley from the centre of the penalty area after Erik Lamela's cross fell to his feet following some shambolic defending by Everton.
The goals were Kane's 20th and 21st of the season in the Premier League -- making it the fifth time he's passed 20 -- and also moved him past Jermain Defoe and Robbie Fowler into sixth for all-time career tallies in England's top flight at 164.
The two goals also saw England captain Kane move two clear of Mohamed Salah's 19 in the race for the Premier League Golden Boot.
Despite the teams playing an entertaining match, neither will likely be happy with the outcome as the single point did not help Everton or Tottenham in their quest for European football next season.
Everton remain in eighth place with 49 points and Spurs, who saw Kane limp off in second-half stoppage time with an apparent injury, are in seventh on 50.