Cornet disclosed that he had conversations with manager Russell Martin and former teammate Flynn Downes regarding his potential move to St. Mary’s Stadium.
"It was a long deadline day but it's finally done and I am happy to be here. I am excited to start working with the club," Cornet said.
"I heard of the interest from Southampton for the first time five days before the end of the transfer window, and I had a chat with the manager.
"The way that he spoke to me was really good and he convinced me to change my club and I am really happy."
The Ivory Coast international added: "I know Flynn from playing together at West Ham, and I know Charlie Taylor from Burnley and we had a good time.
"I asked Flynn about the club and the training ground before I joined. He told me everything is amazing so I said, 'Okay, let's go."