Up against it, the Eagles put their bodies on the line to deny the visitors for 88 minutes and were inches away from a big point until Harry Kane struck.
Spurs had two early chances to take the lead but failed to take them. The first saw a Ben Davies cutback cleared for a corner.
Minutes later a chance fell to Harry Kane after Patrick van Aanholt’s attempted clearance, but the England striker was denied from point-blank range by Wayne Hennessey in the Palace goal.
The first half was spent largely in the Palace half, but Spurs struggled to fashion any clear cut chances against a drilled and determined backline. All of the Eagles’ defensive players were called on throughout the first 45, with debutant Aaron Wan-Bissaka impressing.
It was all square at the break and the second half then continued in a similar vein to the first, Spurs enjoying much of the possession and Palace looking to exploit openings on the counter.
Kane could – and should – have notched the game’s opening goal on 55 minutes, but somehow stabbed the ball wide, unmarked, from five yards out.
Despite their dominance of the ball, Spurs struggled to break through the Palace lines. McArthur, Milivojevic and Riedewald ran themselves into the ground to deny the visitors space, protecting the backline and harrying the opposition.
Serge Aurier had a golden opportunity on 84 minutes, arriving on to a low cross at the back post, but the ball got caught in his feet and he somehow failed to convert. Kane then flashed an effort across goal and wide as Palace weathered the storm.
In the end it was late, late heartbreak for the Eagles. A deep Spurs corner found Kane at the back post and he guided a cushioned header back across the face of goal and into the far side of the net on 88 minutes.
The final minutes passed by without incident, leaving Palace without a point to show for their hard work.
Palace: Hennessey; Wan-Bissaka, Fosu-Mensah, Tomkins (Delaney, 84), Van Aanholt; Milivojevic (c), Riedewald, McArthur; Townsend (Lee, 86), Sørloth; Benteke.
Subs: Henry, Souare, Kirby, Rakip, Lokilo.
Spurs: Lloris (c); Aurier, Sanchez, Dier, Davies; Wanyama, Dembele (Son, 80), Eriksen, Alli, Lamela (Lucas, 66); Kane.
Subs: Vorm, Trippier, Rose, Foyth, Sissoko.