The 22-year-old has long been linked with a move to the south coast and Saints reportedly failed with bids last summer and in January. However, Nigel Adkins now has his man as Rodriguez has signed a four-year deal at St Mary's after leaving the Clarets for an undisclosed fee.
Press Association Sport understands the deal to be worth ?6million, which breaks Southampton's long-standing transfer record, set when they signed Derby's Rory Delap for ?4million in 2001.
"I'm really relieved that we've got it sorted now, and it's done," Rodriguez told the club's official website, www.saintsfc.co.uk. "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.
"As an outsider, you look at the club from where it started, and the drive they have to move up the leagues in quick succession is amazing. The sky is the limit. You see how good the players are at the club. They want to push on even higher, and carry on the way they're going."
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."