Johnson, sacked by Bristol City last month, will replace Jim Gannon to become Peterborough's fourth manager of the season. The 54-year-old has signed a two-year contract.
Former Stockport manager Gannon ended his reign at London Road with a 2-2 draw at Barnsley on Easter Monday, a result which confirmed Peterborough's relegation back to League One.
Johnson, who steered Yeovil into the Football League for the first time in their history in 2003 and to the Coca-Cola League Two title two years later, also won promotion from League One with Bristol City.
The Londoner took charge at Ashton Gate early in the 2005/6 season and guided them into the Coca-Cola Championship the following year. He spent two years as coach of the Latvian national side from 1999 to 2001.
Johnson, whose first game in charge will be at home against Leicester on Saturday, told his new club's official website: "I have had a look around and I am looking forward to getting Peterborough back to the Coca-Cola Championship. I am excited by the challenge.
"I went out to Orlando in America last week and the chairman sold the club to me with his enthusiasm, drive and determination. I am happy to respect anyone who puts money into the football club.
"It is an honour they have given me the opportunity to take Posh back to the Championship. I am relishing the opportunity.
"I came to the game on Saturday against Newcastle and I have about 50 DVDs of matches and I believe the club have the potential to bounce back."
Source: PA
Source: PA