The Three Lions were embarrassed by their visitors at Molineux, paying the price for an abysmal performance that was in keeping with – but altogether worse than – their other disappointing displays this month, which have left England bottom of Group A3 with just two points.
It was England's worst home defeat in 94 years and stretched their winless streak to four, their longest such run since June 2014.
Roland Sallai put Hungary in front early on, but it was not until the latter stages that England's humiliation truly took shape.
Sallai got another in the 70th minute, and that was followed up by Zsolt Nagy's fierce effort and a delicate Daniel Gazdag chip either side of a controversial second yellow card for John Stones.
Defeat in its next game against Italy in September will ensure England is relegated from the top tier of the Nations League, and Southgate fronted up to his mistakes.
"We picked a young team with energy, and when the game started to go against them it started to look that way [that it was a young team], and that's my responsibility in the end," Southgate said.
"I felt at half-time we needed to go for the game, we made changes that gave us a bit more attacking impetus but then we were more open as well, and in the end you're pushing with so many attacking players, that left us wide open.
"I just said to the players there, across the four matches that's my responsibility. I tried to balance looking at new players, tried to rest players, we couldn't keep flogging our more experienced and better players, and in the end the teams I've selected haven't been strong enough to get the results in the two Hungary games really.
"I think the other two performances [against Italy and Germany] have been more positive than perhaps the reaction to them, but I understand tonight is a chastening experience."
Southgate has been a largely popular figure during his reign, but England fans turned on him towards the end of the contest as they chanted "you don't know what you're doing".
Asked if he understood the flak directed towards him, Southgate said: "Of course, in the end this is about winning matches with England, tonight was a night like many of my predecessors have had and experienced, and it's difficult to stand here… I'm not going to say it doesn't hurt, but it's very clear to me what we're trying to do across these four matches.
"The irony is, the two Nations League campaigns are arguably the ones that have heaped negativity and pressure on to us, and you wouldn't normally have that [the Nations League] as an England manager.
"I've got to go with that. I've got to protect the players. They've been exceptional in their attitude, they've never stopped. The results are my responsibility."
But while he claimed to understand the perspective of unhappy fans, he also reminded supporters of his team's past achievements.
"Hungary are a good side, we knew that, and as I said, I think I've given too much for them [England's young players] to do tonight in the end, and I understand the reaction to that in the stadium," he added.
"What I would say is, this group of players has been unbelievable for the country and it's important people stay with them because they're still going to be very strong moving forward."