Nabil Bentaleb opened the scoring in the first half with his first senior goal for the Club, with Nacer Chadli doubling our lead less than a minute into the second period.
Harry Kane then got in on the act, before substitute Roberto Soldado completed the scoring 20 minutes from time.
We will now face Sheffield United in a two-legged semi-final.
Spurs (4-2-3-1): Vorm, Chiriches, Vertonghen, Fazio, Rose, Stambouli, Bentaleb (Capoue 80), Townsend, Chadli (Dembele 61), Eriksen, Kane (Soldado 69). Substitutes (not used): Lloris, Naughton, Kaboul, Lamela.
Newcastle (4-3-3): Alnwick, Dummett, Williamson, Coloccini, Haidara, Cabella (Vuckic 80), Colback, Gouffran, Perez, Sissoko (Anita 74), Riviere (Armstrong 60). Substitutes (not used): Woodman, Streete, Satka, Roberts.
Goals: Spurs – Bentaleb 18, Chadli 46, Kane 63, Soldado 70.
Yellow cards: Newcastle - Haidara.
Referee: Andre Marriner.