Jonathan Woodgate, the former Spurs defender, saw his team take the lead and battle to a 1-1 draw at the Riverside Stadium 10 days ago but they were the architects of their own downfall in north London.
Giovani Lo Celso and Erik Lamela were the respective first-half beneficiaries of poor errors from Boro goalkeeper Tomas Mejias and captain Jonny Howson.
Jose Mourinho's men travel to Southampton in the round of 32, having lost the Premier League version of that fixture 1-0 on New Year's Day, despite Boro substitute George Saville setting up a grandstand finish.
Middlesbrough's hopes of an upset were severely compromised within two minutes when Mejias' loose pass was intercepted by Lo Celso, who capitalised with a composed finish into the bottom-left corner.
Davinson Sanchez overhit his own pass from the back and then lost his footing, allowing Lukas Nmecha to turn inside Japhet Tanganga and draw a fine save from Spurs goalkeeper Paulo Gazzaniga.
Boro were quickly made to regret not capitalising on that opening when Howson lost possession and Lamela eased beyond Paddy McNair and Dael Fry to double the Tottenham lead.
Lamela fizzed over on the turn midway through the first half before Spurs full-backs Ryan Sessegnon and Tanganga went close.
Middlesbrough attacks were becoming increasingly rare, although Ashley Fletcher – their goalscorer in the initial game – could have pulled one back when he headed Marvin Johnson's 44th-minute free-kick wide.
Sessegnon blazed an inviting Tanganga cross over from close range in the 51st minute but Spurs did little else to quicken the pulse for most of the second period.
Saville had a speculative effort deflected on to the roof of the net and Spurs did not heed the warning as he was allowed to advance and slot past Gazzaniga's apologetic dive to prompt a nervy finale.