Rodriguez, who was named in the PFA Championship Team of the Year last season, formally agreed the move during a holiday in Mexico and put pen to paper on a four-year deal today, much to the delight of Adkins.
"Jay is an exciting young English prospect, so I am really happy he is our first signing of the summer," the Southampton manager said.
"There has been a lot of hard work going on behind the scenes on several fronts and we are delighted Jay has signed for us as we get ready for our pre-season schedule.
"He is an exciting 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.
"He has demonstrated a superb attitude and when we have met him it has only confirmed what our research has told us.
"He has a hungry and enthusiastic attitude with a desire to work hard in the positive environment that we have at Southampton where will be able to develop his talent."