Southgate has managed the U21s since 2013 and was previously in charge of Middlesbrough from 2006 to 2009.
After stepping down, former Bolton, West Ham and Sunderland manager Allardyce said “entrapment had won” and his actions were a “silly thing to do”.
Southgate added: “It’s obviously been a difficult situation for the FA but it was important that there was some stability and continuity for everybody.
“So, from my point of view, it was important to step forward and give us the best possible chance to win these games.
“I’m looking forward to the challenge and I’m confident that we can get good results. We have an excellent group of players.”
Meanwhile, Arsene Wenger again committed his future to Arsenal after speculation linking him with the post.
Wenger, 66, said in July he would “never rule it out” but the Frenchman insisted he will remain at Emirates Stadium.
“I am focused 100% on Arsenal and my priority will always be to this club,” Gunners boss Wenger told BT Sport. “I will have to assess how well I do until the end of the season.”
Former England striker Michael Owen believes the full-time job is Southgate’s to lose.
“If he wins all four games it will take a brave man to say ‘let’s change it now’,” said Owen. “Possession is nine tenths of the law, isn’t it?
“I’ve played under lots of managers and don’t get me wrong, they are a very important part of club football and international football, but when you’ve got so many good players it’s more important the players click.
“Eddie Howe could be a good manager. Southgate’s a good man; he could be fantastic. One day someone will click.”
USA boss Jurgen Klinsmann ruled himself out of the running, saying on Twitter there was “no truth to the rumours regarding England”, while Newcastle boss Rafael Benitez said there was “no chance” he would be leaving the Magpies.
Former Chelsea winger Pat Nevin told BBC Radio 5 live the England job would break, rather than make, a manager.
“If you are under 50, it could ruin your career,” he said. “It is like being the Prime Minister – most walk out smiling because they are glad it is over. Expectation is currently low, but that changes quickly.”
BBC chief football writer Phil McNulty
The Arsenal manager’s credentials easily outstrip those of the other candidates being touted as Allardyce’s replacement.
At 66, Wenger might even share Allardyce’s view when he was appointed that his age and experience make him the perfect fit for international management.
Wenger is the perfect next England manager with the ideal credentials and track record if the FA can formulate a plan to somehow attract him to what many now call an impossible job.