The 46-year-old has signed a four-year contract after impressing during a stint as interim manager, and Glenn has hailed the former Middlesbrough boss as "a great ambassador" for the governing body.
"We are delighted to confirm Gareth as England manager. He's obviously somebody we know well but it's his understanding of international football and the development set-up at St George's Park that is important," he told reporters.
"He performed extremely well during the four games he was in temporary charge and he impressed us during a tough interview process.
"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."
Southgate's first game in full charge of England is away to Germany in a friendly match on March 22, before the Three Lions host Lithuania in a World Cup qualifier at Wembley Stadium.