The 22-year-old joins the Saints for a reported £7million and has signed a four-year contract with the club.
Rodriguez told Southampton’s official club website: “I’m really relieved that we’ve got it sorted now and it’s done. I really can’t wait for the start of the season, and the start of pre-season.
“This is a great club, a massive club and they’re where they belong. It’s a squad I want to be a part of and it’s really exciting.”
Southampton manager Nigel Adkins said: “Jay is an exciting young English prospect, so I am really happy he is our first signing of the summer.
“He is an excellent forward thinking player who has already scored a lot of goals at a young age over the past couple of years.
“His characteristics are what we have been looking for.”