The 28-year-old has impressed since joining the Clarets at the start of this season, playing every minute of his side's 11 Premier League games, and an opportunity has arisen at international level after Jordan Henderson, Fabian Delph and Raheem Sterling were all ruled out.
The London-born player stated last week he retained hope of playing for the Three Lions, saying of himself and his Burnley team-mates: "Maybe one day, if we keep the form going, one of us can get in the squad."
But, speaking after being selected, Cork revealed he was stunned by the call-up and hopes to make it a regular occurrence.
"I'm really, really happy," he told Burnley's official website. "I haven't really got the words for it. I'm looking forward to going to meet up.
"We've had a good start to the season and you're probably not going to get many better opportunities to be in the England squad, because we are doing so well.
"Four or five of the lads in the squad probably could have got in at the same time as me, so I feel lucky that I've been given the chance and, hopefully, if we can keep this form up then maybe one or two others can get in as well."
Jake Livermore has already been added to Southgate's squad after Tottenham's Dele Alli, Harry Winks and Harry Kane were all ruled out.