Goal understands that the Gunners have paid £26 million (€30m/$34m) for the 22-year-old, who becomes the club's fourth signing of the summer, following the arrivals of Sokratis Papastathopoulos, Bernd Leno and Stephan Lichtsteiner.
Head coach Unai Emery told the club's official website: “In Lucas Torreira, we have signed a young player who is a very bright talent in the game. A midfielder with great quality, I have enjoyed watching his performances for Sampdoria in the past two seasons, and we all saw him do very well for Uruguay in the World Cup.
"He is a young player already with good experience, but who wants to keep on growing.
"We welcome Lucas to Arsenal and look forward to him joining us soon for pre-season.”
Torreira, who has only won eight caps for his country, featured in every game as Oscar Tabarez's side reached the quarter-finals, where they were beaten 2-0 by semi-finalists France.
The midfielder made a total of 74 appearances for the Serie A side, scoring four goals.
He will wear the No.11 shirt in north London, and will link up with his new team-mates following a post-World Cup break.
Torreira is unlikely to be the Gunners' final signing of the summer, with the club also pursuing a deal for Lorient's Matteo Guendouzi.
Guendouzi is able to play in a defensive midfield role and made 18 appearances in Ligue 2 last season, while the fee is likely to surpass €8m (£7m/$9.4m) plus bonuses.