The Three Lions were humiliated at Molineux and remain without a win in the Nations League, while Southgate was booed and heckled by disgruntled supporters.
Southgate admits that he was "hurt" by the reaction but urged the fans to stick with his squad as the World Cup approaches later this year.
The England boss told Channel 4: "We picked a young team with energy and when the game started to go against them, it started to look that way. That's my responsibility in the end.
"I said to the players, our four matches is my responsibility. I have to balance looking at new players, rest players. We couldn't keep flogging out our experienced players.
"It was an important experience for lots of young players in these games. We've learned a lot in Germany and Italy games. Hungary are a good side and we knew that. I gave them too much to do tonight."
Southgate added: "I understand the reaction in the stadium, what I would say is that in these sets of fixtures, these players have been unbelievable for the country. They have to stay with them.
"Tonight was the sort of night that my predecessors have had. It's important to stand. I'm not going to say it doesn't hurt."
The result was England's worst home defeat for 94 years and Southgate insists that he will be able to cope with whatever criticism comes his way.
He told reporters: "We've broken a huge number of records and tonight we're on the other side of it.
"Tonight's a difficult night. I've watched as a kid and a player, and I recognise those difficult times. But you can't just be at the front when things are good, and not stand up when things get tough.
"I knew the role before I took it. It's happened in ten days, not eleven months. But I've seen other England managers go through that - the great Bobby Robson was, and look how he was judged in the end.
"Football is emotion, and if the flak comes my way, you have to expect that. You can't be England manager and have the run I've had for the last five years without also having some tough times."