Tottenham won 2-1 in the third round replay, with Giovani lo Ceslo opening the scoring in the second minute, before Erik Lamela doubled their lead on the quarter hour.
Both of Spurs goals came following defensive mistakes by Middlesbrough.
Boro pulled a goal back in the final 10 minutes through George Saville but couldnt find an equaliser.
Tottenham will now face Southampton away in the fourth round.
Spurs are currently eighth in the Premier League with boss Jose Mourinho aiming to see them qualify for UEFA Champions League football again via a top four finish.
“In this moment, (FA Cup) is not the priority, Mourinho said after the victory.
The priority is to play a difficult match against Watford in the Premier League. When Southampton comes, it will be the priority.”
Speaking about the win, Mourinho added: “Three-nil was so close so many times. I told my players at half time if we don’t score [to make it] 3-0, then if went 2-1 we would be in trouble and it happened.
Elsewhere, Newcastle also progressed with a 4-1 win over third tier Rochdale as Miguel Almiron scored again.
Shrewsbury Town knocked out Championship club Bristol City 1-0 and will next host Premier League leaders Liverpool in the fourth round.
Coventry City defeated Bristol Rovers 3-0 with Maxime Biamou netting a double.
Reading also moved into the fourth round with a 2-0 win away to Blackpool.