First-half goals from Peter Whittingham and Roger Johnson handed the Coca-Cola Championship side a deserved victory in a week during which they face a trip to the High Court - with the Bluebirds' existence under threat over a £24million debt.
Cardiff boss Dave Jones said: "Honest to God - and you have got to believe me - once tonight is out of the way, we are in tomorrow and we are getting ready for the next league game now because we have a lot of catching up to do.
"But this must give us an incentive now because, man for man down there, they are absolutely elated and they are ecstatic and everything else.
"Someone asked for champagne - for what? We have just got through to the next round.
"We haven't won it, we have not got to the final and we have got to get prepared. Why would we be drinking champagne?
"That's for the staff on the plane going home."
Whittingham blasted the visitors into the lead with just nine minutes gone when he fired expertly past Mark Schwarzer, and Johnson sent the travelling fans into raptures with a 22nd-minute diving header.
Boro rarely looked like addressing the deficit, with Afonso Alves forcing Peter Enckelman into his only real save of the game after 27 minutes and Stewart Downing curling a second-half free-kick inches over the bar.
City looked anything but a club on its knees and Jones insisted the off-field problems had not been a factor for him and his players.
He said: "All I can do is do what I am paid to do, which is the football side, and leave the people who know what they are doing to sort the court case out this week.
"It hasn't had a bearing on us, except that we haven't been able to bring players in. That's the only beating it has had.
"Yes, there are certain things we would have liked during the season, but we couldn't do it until everything was sorted out.
"We have just got on with it. The players know what they have got to do, and they still pick up all their wages, so it has not had an effect on us in that way."