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Southgate, 46, stepped up from his role as England Under-21s manager in September following Sam Allardyce's departure after only 67 days in charge.
He oversaw two wins and two draws as interim boss and has signed a contract reportedly worth up to £2m a year.
"I have thoroughly enjoyed working with the players over these past four games and I think there is huge potential," said Southgate.
"I am extremely proud to be appointed England manager. However, I am also conscious getting the job is one thing, now I want to do the job successfully.
"I'm determined to give everything I have to give the country a team that they're proud of and one that they're going to enjoy watching play and develop."
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Former defender Southgate, who made 57 appearances for England as a player, becomes the fourth permanent manager of the national team in as many years.
The ex-Middlesbrough boss oversaw a comfortable World Cup qualifying win against minnows Malta, before his team laboured to a goalless draw in Slovenia.
He took a significant step towards being appointed full-time manager with a 3-0 win against Scotland in qualifying earlier this month, before his team squandered a 2-0 friendly lead against Spain at Wembley to draw 2-2 four days later.
Before becoming part of the international set-up, Southgate's only managerial experience was his three-year spell at Boro, who were relegated under him in 2009.
Southgate's understanding of international football and the development set-up at St George's Park has proved important, according to Football Association chief executive Martin Glenn.
"Gareth is a great ambassador for what the FA stands for, he's a very good football tactician and a leader, but beneath that he's a winner and that's an important part of the job," he added.
Southgate's first competitive match in charge as permanent manager will be a friendly against Germany in Dortmund on 22 March.
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England defender Gary Cahill: "He's been there. He's been an England player at major tournaments, has the experience and knows how the players are feeling and the pressure you're under in an England shirt. He can relate."
England midfielder Jordan Henderson: "The manager has been brilliant. The players have got on really well with him. He gives responsibility to players and he listens."
Former England defender Danny Mills: "The job he's done is impeccable. He's made big decisions. A lot of people say he's too nice. Yes, he's an absolute gentleman, and very articulate, intelligent, but he's got an edge."
Former Scotland international Pat Nevin after England's win over Scotland: "Why do you have to give him the job now? There is no reason to do it. Wait until the summer and you have all sorts of options, one being Gareth still. England are going through anyway but you might be turning down the opportunity to get Arsene Wenger or someone of that ilk."
Former senior FA executive Adrian Bevington: "I think he is suited to the ebb and flow, the rhythm of international football. The players want to play for him and he is suited to that job and deserves the opportunity."
Southgate was given his first managerial job at Middlesbrough in 2006, when the then 35-year-old defender quit playing to take charge of his former side.
He led the Teesside club for 151 matches between 2006 and 2009, presiding over their relegation from the Premier League before being sacked with the team fourth in the Championship.
And his 28.4% win percentage in the league means he starts the England job with an unwanted record...
After leaving Boro, Southgate joined the Football Association in January 2011 as head of elite development, becoming England Under-21 boss on a three-year contract in August 2013.
Despite breezing through qualifying, the Young Lions failed to progress past the group stage at the 2015 European Championship.
When Hodgson quit in the summer, Southgate ruled himself out of the running to replace him, with Steve Bruce and Allardyce among the interviewees.
This time around it is not known whether anyone else was on the shortlist or even spoken to.
Southgate was interviewed at St George's Park last Monday by a panel including FA chairman Greg Clarke, chief executive Martin Glenn, technical director Dan Ashworth, League Managers' Association chairman Howard Wilkinson and former England defender Graeme le Saux.
Glenn said Southgate "performed extremely well" during the four games he was in temporary charge and "impressed us during a tough interview process".
The only mark against Southgate's time as interim manager was the FA banning England players from having nights out on international duty after captain Wayne Rooney was pictured drinking at a wedding.
Rooney apologised for the "inappropriate" images - taken on the evening after the Scotland game at the team hotel - while the FA was looking into reports that other squad members were out late three days before the Spain friendly.
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Former England midfielder John Salako, who played with Southgate at Crystal Palace, speaking to the BBC.
Gareth had an inner steel, an inner confidence. Captain Sensible we called him. First out to training, last off the training pitch. He was a bright, intelligent lad. Not the most gifted of players but he worked on everything so hard.
Gareth was a real student of the game - he always had that aura about him that he could go on to be whatever he wanted to be.
Getting arguably the biggest job in football, I perhaps didn't see that coming. But the opportunity has come along. We're in turmoil a little bit. Gareth's a safe pair of hands and I'm thinking why not? He's won the Toulon tournament, he knows the crop of youngsters coming through, he knows what English football is about.
We want an Englishman in charge who was proud to put on that shirt, who knew what it was about, who had the passion to go and play for England. And he'll do that as England manager.
Ok he hasn't managed at the top Premier League level, Champions League level, but I don't think that matters. Get behind him, give him the opportunity to go and be a great English manager.
It is the perfect time for Gareth to step in. He'll do it from the heart, will use his head. No ego in it. He's not in it for the money, it's more about the prize of being England manager.
Phil McNulty, chief football writer
Southgate has rebuilt his managerial career since taking over as England Under-21 coach in August 2013. He endured a miserable Euros in 2015 when England's youngsters went out at the group stage but he was hampered by the FA's decision to omit players such as Raheem Sterling, Jack Wilshere and Ross Barkley.
He saw the other side as England won this summer's prestigious Toulon tournament in France - but is this enough to warrant the top job?
Southgate is steeped in FA tradition, having had an 18-month spell as its head of elite development between January 2011 and July 2012 before his return as England Under-21 boss.
He knows what the England job entails after close contact with the likes of Roy Hodgson and Sam Allardyce and has clearly developed a taste for the pressures after his time in interim charge.
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Gareth Southgate has been appointed as England manager on a four-year deal.