Adkins, 47, took the club from League One to the Premier League with successive promotions.
Pochettino, 40, will take charge against Everton on Monday.
The Argentine left Spanish side Espanyol in November and will lead training on Saturday for the first time.
Southampton chairman Nicola Cortese
said in a statement:
"This decision has been made with the long-term ambitions of Southampton Football Club in mind.
Adkins' Southampton record
"Whilst we acknowledge the contribution Nigel has made during the past two years, for the club to progress and achieve our long-term targets a change was needed.
"Mauricio is a well-respected coach of substantial quality who has gained a reputation as an astute tactician and excellent man manager.
"I have every confidence that he will inspire our talented squad of players to perform at the highest possible level."
Pochettino played for Espanyol and Paris Saint-Germain as a defender during his career, and won 20 caps for Argentina, fouling Michael Owen at the 2002 World Cup to give England a penalty which was converted by David Beckham.
Pochettino said: "This is the kind of opportunity that any coach would relish. Southampton is a club with great heritage, and an even more exciting future.
Adkins' Saints march up to Premier League
"There is a clear vision to take the club to a new era of sustained success in the Premier League, and beyond, which I'm delighted to be part of."
Pochettino took charge of Espanyol at the age of 36 in January 2009, and managed a best finish of eighth in La Liga in 2010-11.
He left the club by mutual consent at the end of November 2012, with an overall record of 49 wins from 146 league matches.
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