The 51-year-old will replace Jorginho, who was sacked after only 14 matches, becoming the third coach this year at the club.
"I'm very proud," Menezes wrote on his Twitter account.
Menezes was in charge of Brazil for just over two years, during which he painstakingly rebuilt the team and moved them away from their unpopular counter-attacking game towards a more possession-based style.
He was surprisingly fired just when he seemed to have found his ideal team after two years of experimenting.
Menezes admitted that five-times world champions Brazil had been left standing by Spain and shortly before his dismissal said they would only catch up if they adopted a long-term approach.
Menezes made his name in 2005 when he led Gremio out of the second division, clinching promotion in an extraordinary game when Gremio had four players sent off, survived a penalty miss by opponents Nautico and then snatched a goal to win 1-0.
Two years later, he took Gremio to the final of the Libertadores Cup, the South American equivalent of Europe's Champions League. He then joined Corinthians, another hugely popular team, and led them out of the second division in 2008.
Flamengo are one of Brazil's most popular clubs but have been dogged by financial problems and unpredictable leadership.
They have won one of their first five games in the Brazilian championship and are 14th in the 20-team table.
