England suffered their worst home defeat for 94 years when they lost 4-0 to Hungary earlier this month, with Southgate booed by fans at Molineux.
An away defeat against Hungary and draws against Germany and Italy led to fierce criticism of Southgate just five months out from the start of the World Cup in Qatar.
But the FA have given the former Middlesbrough boss their backing. Southgate, who led England to the Euro 2020 final on home soil 12 months ago and the World Cup semi-finals in Russia in 2018, signed a two-year contract extension last November.
And FA chair Debbie Hewitt said on Tuesday: “My personal opinion on Gareth is that he is, by the facts on the pitch, the most successful England manager we've had for 55 years. The bit people don't see as much is the Gareth at camp and the culture he's created.
“Certainly prior to Gareth being the manager of England, there was not the pride of wearing the England shirt. There were the club rivalries we'd read about. The players not getting on. He's changed that beyond recognition and I've seen that first hand.
“I'd also say that I don't just work in football, I work in business and I've worked with a lot of chief executives and Gareth's skills - his high IQ and high EQ - would make him a chief exec in any sphere.
“That resilience and accountability (are) the two qualities I admire most. There are no slopey shoulders, he doesn't huff, he's resilient and that's what you want in an England manager."