The Three Lions suffered their worst home defeat since 1928 to a team 40 places below them in the FIFA world rankings - and fans started to turn on Southgate.
Chants of 'you don't know what you're doing' and 'sacked in the morning' rang around Molineux as England failed to win for the fourth consecutive Nations League match.
The backlash from supporters left talkSPORT correspondent Faye Curruthers no choice but to ask captain Kane about Southgate's future - and the striker delivered a defiant response.
"It's a really disappointing question to be answering," Kane told talkSPORT.
"If you look at where we were at Euro 2016 and where we are now, we've taken massive strides and Gareth has been a massive part of that.
"It's disappointing. I understand it's football. I understand the fans want answers. But he's been England's most successful manager in a long time.
"We're fully behind him. We've got a process that we'll stick to and we know. We're not going to panic."
Kane admitted it was a night to forget for England, but insists they will not overreact.
He added: "It's a really disappointing night, there's no beating around the bush. A night to forget, for sure.
"It's really disappointing when you're in front of your home fans and you want to try and put on a show and win.
"We will process it and look at it but sometimes football throws these games up out the blue and we have to take it on the chin.
"There's no hiding from it, but we'll use it to prepare for another major tournament this year - and that's go to be the bigger picture."
Meanwhile, England legend Stuart Pearce was among those to back Southgate after criticism from a large section of the home faithful in Wolverhampton.
Many have been quick to highlight Southgate's prolonged success as England boss, taking the Three Lions to the semi-finals of the 2018 World Cup and the final of Euro 2020.
But Pearce insists that while the last four results do not call for a change of management, they certainly should not be dismissed.
He said: "It's really difficult to defend and I'm on of the most positive England fans.
"I didn't see this result coming. We, as a squad, have probably got to go and a good look at themselves.
"As we stand, in this block of four games, our performances have been woefully short of where we need to be.
"But changing the manager? For me, it's a no."