The Brazilian settled the all-Premier League fourth-round clash in the 68th minute at Turf Moor, firmly planting home a header from Emerson Royal’s cross.
Victory means the 2021 finalists of this competition are through to the last eight for the third time in four years.
Nuno Espirito Santo's side also registered a first clean sheet in 11 attempts, and only a second away win in eight.
Tottenham had only lost once in their 14 previous meetings with Burnley and had a good chance after just three minutes.
Giovani Lo Celso played Harry Kane through on goal, but the England captain uncharacteristically dragged his effort wide.
Emerson came to Spurs' rescue at the other end 10 minutes later; the full-back importantly clearing Johann Gudmundsson’s deep cross with Dwight McNeil lurking.
Aiming to beat a Premier League opponent over 90 minutes for the first time in five attempts in this competition, Spurs turned up the pressure after the break.
Nick Pope produced brilliant reflexes to deny Lo Celso from Kane's through ball in the 65th minute.
However, the Burnley goalkeeper was powerless three minutes later as an unmarked Lucas – on as an early substitute for the injured Bryan Gil – headed in Emerson's cross from close range.
The hosts pushed for an equaliser late on with Ashley Barnes testing Pierluigi Gollini from distance.
However, Spurs held on and the Clarets' wait for a first quarter-final appearance since the 2008-2009 season continues.