We fell behind on the half-hour mark when Jan Vertonghen accidentally turned the ball in at the near post from Wilfried Zaha's cross, but Harry Kane restored parity with a powerful header on 63 minutes.
Vertonghen was unfortunately forced off injured with Kevin Wimmer coming on in his place for his Premier League debut with 15 minutes remaining.
Then came a moment of magic from Dele Alli as he scored a marvellous goal to put as ahead on 84 minutes, before Nacer Chadli found the net in stoppage time with a long-range strike.
Crystal Palace (4-2-3-1): Hennessey, Ward, Dann, Delaney (c), Souare; Ledley, McArthur (Jedinak 78); Puncheon, Cabaye (Campbell 85), Zaha; Wickham. Substitutes (not used): Speroni, Lee, Mutch, Chamakh, Kelly.
Spurs (4-2-3-1): Lloris (c), Trippier, Alderweireld, Vertonghen (Wimmer 75), Rose; Dembele, Dier (Chadli 56); Eriksen (Bentaleb 88), Alli, Son; Kane. Substitutes (not used): Vorm, Davies, Carroll, Lamela.
Goals: Crystal Palace – Vertonghen OG 30. Spurs – Kane 63, Alli 84, Chadli 90+5.
Yellow cards: Crystal Palace – McArthur 60, Jedinak 90+3. Spurs – Trippier 79, Kane 90+3.
Referee: Martin Atkinson.