"It's the first derby I've lost as a manager, so that's painful," Southgate said after the match.
"You don't want to be in the middle of somebody else's party when you've lost.
"But sometimes as a group, you're going to have to go through these experiences and how you come out the other side is the most important thing.
"It's fantastic for Sunderland that they're secure, and deservedly so with what they've done during the season.
"Hopefully we can have a similar thing next week."