Kane scored in the 88th minute, after missing an earlier chance.
Wayne Hennessey made a number of fine stops to keep the North Londoners at bay and Kane spurned a couple of decent opportunities.
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.
Referee Friend did point to the spot on the half-hour mark after Timothy Fosu-Mensah brought down Kane, only for the official to quickly change his decision with the assistant referee's flag correctly raised for offside.
The Cote d'Ivoire global was involved at the other end shortly after, doing brilliantly to deny Alexander Sorloth.
Ben Davies should have done better after Andros Townsend unexpectedly failed to clear Eriksen's free-kick although the left-back insisted he had a strong claim for a penalty after colliding with Hennessey.
Palace's first chance arrived two minutes later as Tomkins drew a routine save out of Hugo Lloris with a looping header from a free-kick on the left, before the French keeper was called into action again on 16 minutes to prevent Christian Benteke from getting on the end of Van Aanholt's cross from the left.
"I'm so happy. Three points".
And Tottenham. We haven't even come to them yet - they could well profit nicely with a victory at struggling Crystal Palace.
Lucas Moura could make his full Premier League debut, having started against Rochdale in the FA Cup. The Spurs are in a heated battle for a spot in the Champions League meaning that every point counts.