Spurs were precocious a second-minute lead once Boro goalkeeper Tomas Mejias passed the ball to Giovani lo Celso, who cut within a challenge and scored.
Erik Lamela doubled the hosts’ lead after fifteen minutees once he flicked the ball past Mejias after a fine run.
George Saville force one back late on for Boro with a low strike from twenty yards out, however it had been not enough.
Middlesbrough substitute Rudy Gestede had an opportunity to force extra-time however he could solely head over the bar from eight yards as Spurs endured for the win.
Tottenham, who have won the fa Cup eight times, will play at Southampton within the fourth round on twenty five January.