The 22-year-old netted 21 goals in 42 games for the Clarets in all competitions last season, and is widely regarded as one of England’s most exciting attacking prospects.
It is a case of third time lucky for the south coast club, after they had bids for the once-capped England Under-21 international rejected in the last two transfer windows.
Rodriguez expressed his delight at finally sealing the move on the club’s official website, saying: "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.
"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.
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"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 partnered Saints striker Rickie Lambert up front in the Championship Team of the Year, and manager Nigel Adkins can now look forward to seeing the two link up for real at St Mary’s next year.
The former Scunthorpe boss was over the moon after tying his new signing down to a four-year deal, saying: "Jay is an exciting young English prospect, so I am really happy he is our first signing of the summer.
"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."