Boro – managed by Jonathan Woodgate, who headed Spurs to victory in the 2008 League Cup final – took a 50th-minute lead through Ashley Fletcher's sixth goal in 11 games.
But Jose Mourinho's men hit back within 11 minutes when Brazilian Moura headed firmly past Tomas Mejias to secure a 1-1 draw and a place in Monday evening's fourth-round draw.
Tottenham were well worth a replay on the balance of play on a day when the injured Harry Kane's absence blunted their attack, but Boro equally deserved another crack after a spirited display against last season's Champions League finalists.
Going into the game on the back of four successive Championship wins, the Teessiders started confidently, although it was the visitors who mustered the first attempt on target when Christian Eriksen fired in an eighth-minute shot which goalkeeper Mejias could not hold.
Mejias had to race from his line to deny Son Heung-min two minutes later after Moura embarked on a mazy run before sliding the ball into his path.
Spurs nearly took a 20th-minute lead when Boro defender Paddy McNair diverted Eriksen's cross just past a post as he attempted to block it.
But they were almost behind three minutes later as keeper Paulo Gazzaniga had to pull off a double save to deny first Dael Fry and then George Saville from McNair's deep free-kick.
With Son and Moura linking well with Eriksen and Dele Alli – the England international was harshly booked for diving – and wing-backs Serge Aurier and Ryan Sessegnon repeatedly finding space, Spurs continued to look the more threatening as the half-time whistle approached, although without causing Mejias undue problems.
But it was the home side who took the lead five minutes into the second half when Saville, who had seconds earlier seen penalty appeals waved away, clipped the ball over the top of a flat-footed Tottenham rearguard for Fletcher to race clear and beat Gazzaniga.
Mourinho's response was to replace Sessegnon and Harry Winks with Giovani Lo Celso and Erik Lamela, but the visitors continued to lack the precision they needed in the final third to drag themselves back into the tie.
However, they did just that in the 61st minute when Aurier drilled a cross to the far post and Moura got between Jonny Howson and Djed Spence to head home an equaliser.
The game opened up as both sides went in search of a winner and Fletcher might have done better when he shot straight at Gazzaniga with 20 minutes remaining.
Mejias just about managed to turn Lamela 74th-minute strike wide as it skipped nastily off the turf, and then pulled off a fine save to keep out Moura's piledriver with nine minutes left.