Palace had introduced Damien Delaney for the injured James Tomkins but the Irishman lost track of Kane in the penalty area, and the forward met Eriksen's corner to steer a header towards goal which Hennessey failed to keep out despite getting a strong hand to the ball.
Tottenham boss Mauricio Pochettino explained: "It will be a big battle".
"We left that very late", Kane said.More news: Onus on Mourinho to get Pogba performing for Conte clash
Palace were fortunate not to concede a penalty inside eight minutes - Hennessey's indecision nearly proving costly when he brought down Ben Davies, but Kevin Friend waved play on.
Wayne Hennessey had saved an early effort from eight yards before Kane missed past the post with the whole goal at his mercy from six yards in the second half. Referee Kevin Friend pointed to the penalty spot, but the linesman ruled the England striker offside. The home side grew in confidence, keeping their defence compact making it hard for Serge Aurier and Christian Eriksen on the right to find any pockets.
The Ivory Coast global was involved at the other end shortly after, doing brilliantly to deny Alexander Sorloth. It also marked Tottenham midfielder Erik Lamela's 100th Premier League appearance. Eriksen dinked a lovely cross that dropped right to the feet of Kane around two-yards out, but he directed his shot just wide.More news: Man City's Pep Guardiola to wear yellow ribbon at Carabao Cup final
We're in for a cracking Super Sunday and it kicks off with Crystal Palace v Tottenham at Selhurst Park at midday. His 150th goal for his club.
- Kane has netted 23 goals in 34 Premier League London derbies - equalling Didier Drogba's total in the competition (23 in 64).
Roy Hodgson is without 12 players for Crystal Palace as they suffer an injury crisis.More news: Trump also met the Mexican president once during the 2016 election campaign
He joined The Eagles from Leicester City in January previous year.