Southgate admitted the Teessiders were ripe for an upset against the Coca-Cola Championship leaders.
Boro arrived in the Black Country with only one win from their last 11 matches - a slender 2-1 victory at home to non-league Barrow in the previous round.
But their recent form in this competition has been good - they have reached the sixth round in each of the last two seasons and were losing semi-finalists in 2006.
They progressed again courtesy of a deft finish from Afonso Alves on the stroke of half-time and a fortunate late strike by substitute Marvin Emnes.
Southgate said: "I think it is important to win because in the last two years we have made the quarter-finals twice, and it has coincided with our league form improving.
"A lot of people expected us to get turned over and that it would be hard for us to get a result because they have got momentum and a league position and we haven't got that at the moment. But that didn't happen.
"It was a big ask as we had a young side out there but they responded to everything that was thrown at them and that is great credit to the group of players."
Goals have also been a problem for Boro recently - they have managed just three in their last eight league games.
But they created a host of chances at Molineux. The problem was that most of the best ones seemed to fall to Adam Johnson and not one of their more recognised goal-scorers.
"At least we are creating chances," said Southgate. "We had umpteen in the first half and could have been out of sight.
"There was some really good attacking play from us. The first goal in particular was a great move and we also had a couple of moves similar to that.
"That will give the lads confidence and, of course, hitting the back of the net is something we have not done often enough of late, but what is pleasing is that we are still creating chances.
"Adam (Johnson) could have had a hat-trick but he kept getting in there. He was there to miss them and that is the pleasing thing."