The 27-year-old Italy international has long been linked with a move to the Premier League, with Manchester City and Tottenham reportedly interested.
Osvaldo worked with Southampton manager Mauricio Pochettino when the pair were at Espanyol, although getting him to the south coast took some persuasion.
The Argentinian-born forward, though, has now penned a four-year deal at St Mary's, with the fee set to be confirmed to the Italian stock exchange by Roma on Monday.
"Southampton Football Club is delighted to announce the signing of Italy striker Pablo Daniel Osvaldo," a statement on the club's official website, www.saintsfc.co.uk, said.
"Osvaldo today agreed terms on a four-year contract with Saints and moves from Serie A side AS Roma. The frontman is expected to be available for next Saturday's Barclays Premier League game against Sunderland at St Mary's, and will be officially presented as a Southampton player tomorrow (Monday)."
Osvaldo becomes Southampton's third signing of the summer, following an £8.5million move for Lyon defender Dejan Lovren and the signing of Kenyan midfielder Wanyama.
Southampton won their opening match of the season at West Brom on Saturday , with a late Rickie Lambert penalty securing a 1-0 victory.
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