The eight-time champions trailed to Ashley Fletcher’s fine finish just after the break but Moura’s header spared a Tottenham side who were without injured striker Harry Kane.
Tottenham dominated following Moura’s equaliser but Boro keeper Tomas Mejias, making his first start of the season, frustrated them with several saves as Boro earned a replay.
A much-changed Chelsea booked their place in round four with a comfortable 2-0 win over Nottingham Forest who they also beat at the same stage of the competition last season.
Callum Hudson-Odoi and Ross Barkley were both on target for eight-times winners Chelsea in the first half.
Former Manchester United and England striker Wayne Rooney helped Championship side Derby County spring a surprise with a 1-0 victory at Premier League Crystal Palace.
Top-flight Sheffield United were made to work hard by minor league outfit AFC Fylde, winning 2-1.
In the day’s later kickoffs runaway Premier League leaders Liverpool host Everton in the Merseyside derby while West Ham United travel to Gillingham.