Southgate, who replaced Sam Allardyce after his short-lived spell, won 61 of his 102 games in charge of England, drawing 24 and losing 17.
The former defender, who won 57 caps for England between 1995 and 2004, has been involved in the England set-up since 2013, having managed the under-21s for three years prior to taking the top job.
His only experience in club management was at Middlesbrough from 2006-2009.
Southgate thanked his players and backroom staff, calling assistant Steve Holland “one of the most talented coaches of his generation”.
“I joined the FA in 2011, determined to improve English football,” he added.
“I hope we get behind the players and the team at St George’s Park and the FA who strive every day to improve English football, and understand the power football has to drive positive change.”
Bullingham said Southgate had made “the impossible job possible and laid strong foundations for future success”.
“We are very proud of everything Gareth and Steve achieved for England, and will be forever grateful to them,” added Bullingham.
“Over the last eight years they have transformed the England men’s team, delivering unforgettable memories for everyone who loves the Three Lions.
“We look back at Gareth’s tenure with huge pride – his contribution to the English game, including a significant role in player development, and in culture transformation has been unique.”
