Allardyce was sacked by the Football Association (FA) after just one game in charge for behaving "inappropriately" as he sought a lucrative sideline role in an undercover sting set up by the Daily Telegraph newspaper.
"It's obviously been a difficult situation for the FA but it was important that there was some stability and continuity for everybody," Southgate told the FA's website (www.thefa.com).
"So, from my point of view, it was important to step forward and give us the best possible chance to win these games.
"With my experience with the under-21s in international football, and having worked with so many of the players who have moved up to the seniors as well, I think we are in a good position to prepare for these games."
The former England centre-half ruled himself out of the running to take charge of the senior side in July when Roy Hodgson resigned following England's dismal Euro 2016 campaign but Southgate is looking forward to the challenge ahead.
"I was very proud to have played more than 50 times for my country and to have worn the armband, so to manage England will be a huge responsibility and a huge honour for me," the 46-year-old added.
"It's a difficult circumstance for me, because we've lost two senior coaches in the last few months who have both been very good to me, so that complicates my feelings. I certainly wish them both all the best for the future."
Southgate will also lead England against Scotland in a third World Cup qualifier in November and in an international friendly against Spain.
(Reporting by Ian Rodricks in Bengaluru)
