First-half goals from Peter Whittingham and Roger Johnson handed Cardiff side a deserved victory in front of a full-house at Boro and a dejected Southgate said afterwards: "It is a huge disappointment for us. We know what an opportunity it was to win something.
"But that is the cup and that is why I always say you are judged as a team by what you do in the league because the cup can go any way for you.
"We have had games here that we should have been out of competitions and we have gone all the way to the final.
"Today, we got what we deserved. It's what happens on the day and we didn't deliver."