Connor Ogilvie struck twice at Leyton Orient's Matchroom Stadium to secure our passage into round four.
The defender opened the scoring on 28 minutes with a header from Harry Winks' corner before curling home a fine free-kick in the 70th minute. Boro responded via Bryn Morris with six minutes remaining.
Spurs: McGee, Walker-Peters, Ogilvie, Winks, Carter-Vickers, Lesniak, Sonupe (Georgiou 78), Onomah, Harrison (Akindayini 81), Miller, Oduwa. Subs not used: Vincent-Young, Pritchard, Priestley (GK).