After a frustrating first half, we took the lead on 56 minutes when Nacer Chadli fired home his ninth goal of the season.
Burnley fought back, however, and equalised through substitute Sam Vokes on 73 minutes - forcing a replay at White Hart Lane next week.
Spurs (4-2-3-1): Vorm; Chiriches, Fazio, Vertonghen (c), Davies; Dembele, Stambouli (Kane, 46); Chadli, Paulinho, Eriksen; Soldado (Townsend, 67). Substitutes not used: Friedel, Dier, Naughton, Capoue, Onomah.
Burnley (4-4-2): Heaton; Trippier, Keane, Mee, Lafferty; Kightly, Arfield, Marney (c), Boyd (Wallace, 78); Barnes (Vokes, 61), Ings (Sordell, 78). Substitutes not used: Gilks, O’Neill, Reid, Jutkiewicz.
Goals: Burnley - Vokes, 73; Spurs - Chadli, 56.
Yellow cards: Spurs - Stambouli, 10.
Referee: Roger East.