After a stalemate against Sheffield United on the opening day of the campaign, Boro arrived at the City Ground on the back of a three-game winning run and yet to concede a goal this season.
Luke Chambers ended that streak after an hour, cancelling out Adam Johnson's opener two minutes before the break.
Radoslaw Majewski then struck towards the end of the first period of extra time to send Forest into the third round of the cup for the first time in five years.
Boro travel to Ashton Gate to face Bristol City at the weekend and Southgate said: "We've got to move on very quickly and make sure we show we can react to a defeat because Bristol City are one of the teams that are going to be up there or thereabouts.
"A record like that is going to go at some point and I would rather it be tonight than Saturday.
"We wanted to have a go and that's why we fielded a strong team because we wanted to maintain that momentum.
"I felt we did that because we started for an hour very well - it was as good as we've passed the ball all season.
"We created good chances and took our goal well, it was a really good move and well finished."
On the speculation surrounding his star players, Southgate reported no new developments.
Tuncay was left out with a knock and Southgate revealed there was interest in the striker both at home and on the continent.
He added: "There's nothing concrete on any of our players, there's a lot of noise from outside but nothing confirmed on anyone at the moment.
"Tuncay had a knock. He's one who there is a bit of interest in - in this country and abroad - at the moment and we will have to see what develops."
Forest boss Billy Davies hailed the fitness of his players, but the Scot is looking forward to the forthcoming international break so his squad can rest up and recover from injuries.
"I've said for weeks and weeks that I have been pleased with the attitude and spirit of the players and the general fitness of the side," said Davies.
"That was evident again and Middlesbrough are a top team who have been working together for a long time, and that's what we need with this team - time.
"We need time to get over these consecutive games, time to get one or two guys back from injury and time to work with individuals on certain things and if we can do that we will get better.
"I've been proud of the players' performances since the start of the season - the Watford game here last week aside.
"It will be nice to get to the international break and rest these guys. Some of them have played nine games in a month and that is way too much."