The 45-year-old has agreed a three-year contract and will arrive at St Mary's next week to prepare for the new season.
Pellegrino, who won La Liga as a defender with Valencia, began his coaching career as an assistant at Liverpool before spells with Valencia, Estudiantes, Independiente and Alaves.
"I am delighted that after a diligent selection process we have recruited Mauricio Pellegrino as our new first team manager," Southampton vice chairman Les Reed said in a club statement.
"We had an impressive list of candidates and went through a comprehensive interview process. Mauricio impressed us throughout with his depth of knowledge, motivation and ability, communicating this in a professional and passionate way.
"He has an excellent understanding of the Southampton way, and his style of play and aspiration matches the philosophy, culture and ambition of the club."
It is the second time Southampton have turned to an Argentine coach having employed Mauricio Pochettino for 14 months before he joined Tottenham Hotspur in 2014.
Pellegrino stepped down as Alaves manager last month after leading them to the Spanish King's Cup final and finishing ninth in La Liga with a team who had only been promoted to the top flight the previous season.
He said he was "excited and passionate" about the future with Southampton.
"I think my philosophy and the culture of the club will work well together," he said.
"We have to be a team on and off the pitch; I want to make the supporters proud of the players and to feel part of the game every week."
Frenchman Puel was fired after less than a year in charge despite taking Southampton to the League Cup final and eighth place in the Premier League table.
Southampton won only one of their final eight league games, scoring just one goal in their last five home matches, and ended with their lowest points tally in four years.
(Reporting by Martyn Herman, editing by Ed Osmond)
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