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They had come from 2-0 down to earn a draw in the original fourth round tie at Ashton Gate, and stunned their illustrious hosts by taking a first-half lead in the replay through David Noble.
Australia international Mark Viduka was brought on as a half-time sub by Boro manager Gareth Southgate, and he equalised to send the tie into extra-time.
They should have taken a 2-1 lead when City captain Jamie McCombe - a former Scunthorpe player - conceded a penalty kick.
But Aiyegbeni Yakukbu, who had scored 17 of his previous 18 spot kicks, was outfoxed by visiting keeper Chris Weale.
Weale was making only his second-ever appearance for the Robins, having made his bow only on Saturday after No.1 keeper Adriano Basso was sent off in a league clash.
Weale did not dive, knowing Yakubu takes a slow run-up, and stood still to make an easy save.
Yakubu made up for his miss by giving Boro the lead, but although they pushed for a third goal were hit by a sucker punch as McCombe grabbed a last-gasp equaliser.
City's Phil Jevons missed the opening penalty, and a succession of conversions appeared to have Boro into the next round.
But there was still time for more late drama as, incredibly, Yakubu missed another spot-kick.
He struck the outside of the post with a kick which would have sealed the win, but City's Scott Woodman saw his next effort saved leaving Adam Johnson to - finally - send Boro up for a fifth round game against West Bromwich this weekend.
Blackpool went out after a 3-2 extra-time defeat at Norwich, who will now face Chelsea in the fifth round.
The Seasiders took a first-half lead through Michael Jackson, but Darren Huckerby levelled to send the tie into extra-time.
Huckerby struck again five minutes into the extra period, and although Shaun Barker thought he'd earned the Lancashire side a replay until Chris Martin grabbed the winner.
There was one League One game played on Tuesday. Brighton eased their relegation worries with the 4-1 demolition of dismal Leyton Orient, who remain stuck firmly in the drop zone.