"We had plenty of chances to win it, but the longer it went on the more it looked like it was going to penalties," Southgate said after the match at the Riverside.
"But we have shown a lot of heart and character this season and we showed that again. Everyone left in the last eight of the FA Cup will think they are in with a great chance.
"The players were on their last legs but they have pulled through again. It was a bizarre goal and a bit fortunate, but we thought Mido's shot might have been handled by a defender [Chris Morgan] before the ball ended in the net."