At half-time, after goals by Nigerian striker Yakubu and Malcolm Christie - standing in for hamstring victim Mark Viduka - the Premiership side appeared to have all but booked their passage into round five.
But falling behind is something the Robins know all about at League One level. It has happened to them six times in the last seven games and in two of them they were also two goals down.
Their manager, Gary Johnson, was able to build on this recent history at the break as he fired his side up by telling them to take the game to Boro.
Even Gareth Southgate admitted at times his side, surprised by the strength of City's counter-attacks, were hanging on and being outplayed.
Hang on they did but it was two free-kicks, both 35 yards out, which landed them in trouble.
Both were given away by 18-year-old Lee Cattermole for challenges on Lee Johnson, son of the City manager, and Scott Murray.
In the first half Cattermole had got away with a tackle from behind on Johnson, but now referee Graham Poll was unforgiving.
Johnson swung the ball in and Irishman Richard Keogh rose to head the ball past Mark Schwarzer.
The second half was just eight minutes old when that goal went in, but five minutes later City struck again, also from the left side, but slightly deeper.
Johnson's kick was identical, Schwarzer came off his line to punch away but, from 30 yards, Murray, deep on the right, volleyed the ball home to register his 88th goal for the club.
Boro were all over the place and for once even Jonathan Woodgate looked uneasy as City, who had started the game with only one forward, in Enoch Showunmi, began chasing every possible chance.
There was so nearly a goal for Jamie McAllister as the sell-out crowd of 19,000 found their voice.
Showunmi breasted-down a chance to David Noble, which was hit over the top and Cole Skuse saw his shot kicked off the line by Stewart Downing.
It had all been so different at the start when Middlesbrough pounced on some sloppy passes and played square along City's back line to seize their goals.
Louis Carey, so often solid in City's defence, was slow to react in the fourth minute, as a ball directed towards him was about to fall short. The gap was there and Downing and Yakubu were in the clear, but still needed a little help from Brazilian keeper Adriano Basso.
He spilled Downing's close-range shot but Yakubu was there to fire the loose ball high into the net.
City learned nothing from this and 20 minutes later another square ball across defenders, this one from Liam Fontaine, brought the second goal, neatly turned in by Christie, who had been called into the side when Viduka reported a slight hamstring pull.
City had gone into the game with Showunmi as a sole striker, with five across the midfield.
The plan was for Noble to fill in the holes and draw away the Middlesbrough centre-halves, but City never found the pace and freedom of thought to worry their visitors until their manager pointed out the error of their ways at half-time.