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 longstanding 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 his new club's official website. "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."