Jose Mourinho's side has been in erratic form in the Premier League and it showed little improvement at the Riverside Stadium, where Ashley Fletcher scored his fourth goal in five games to give Boro the lead early in the second half.
Boro boss Jonathan Woodgate's hopes of claiming a Premier League scalp against one of his former teams were dented as Lucas headed Spurs back on terms 11 minutes later, but the Championship club's supporters will not begrudge him a grand day out at Tottenham Hotspur Stadium in the replay.
A troublesome afternoon for Mourinho saw midfielder Harry Winks suffer a suspected ankle injury, but it is defensive lapses and an apparent lack of confidence that should most concern the Spurs boss.
Christian Eriksen kept his place in the Tottenham team amid reports linking him with a transfer to Inter and he went close to scoring after seven minutes, his shot from 20 yards hitting Boro keeper Tomas Mejias in the chest.
Tottenham was on top until Dael Fry's powerful downward header produced a goal-line save by Paulo Gazzaniga, Middlesbrough's counter-attacks proving more threatening than Spurs' periods of pressure against a packed home defence.
The second half was five minutes old when George Saville's chipped pass put Fletcher in behind the Tottenham defence and the former West Ham forward clipped the ball over Gazzaniga, who got a hand to it but could not keep it out.
However, the equaliser provided a reminder of Boro's defensive frailties, Serge Aurier crossing to the far post where the diminutive figure of Lucas was allowed to head into the net unchallenged.
Fletcher stung Gazzaniga's palms with a side-footed drive from the edge of the box as Spurs wobbled in the last 20 minutes, while substitute Erik Lamela and Lucas both had shots well saved by Mejias at the other end, but neither side could force a winner in a tense finish.