Dunga was the captain of Brazil's 1994 World Cup-winning team and coached the national team from 2006 to 2011.
"I am immensely happy to be back," said the 50-year-old, who was the favourite to land the job.
Dunga replaces Luiz Felipe Scolari, who resigned after Brazil reached the semi-finals of the 2014 World Cup, but suffered the joint-heaviest in their history, losing 7-1 to Germany.
Dunga was in charge at the 2010 World Cup in South Africa when Brazil were knocked out in the quarter-finals.
Meanwhile, Argentina coach Alejandro Sabella, 59, will decide his future next week after his side lost 1-0 to Germany in the World Cup final.
Members of the squad have said they want Sabella to remain in charge for the 2015 Copa America in Chile.
"It's down to whether I feel strong enough to give 100%," said Sabella.
"I am immensely happy to be back," said the 50-year-old, who was the favourite to land the job.
Dunga replaces Luiz Felipe Scolari, who resigned after Brazil reached the semi-finals of the 2014 World Cup, but suffered the joint-heaviest in their history, losing 7-1 to Germany.
Dunga was in charge at the 2010 World Cup in South Africa when Brazil were knocked out in the quarter-finals.
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Scolari, who guided Brazil to World Cup victory in 2002, and his entire backroom staff have all
left their posts
since the tournament finished.
Defeat worst day of my life - Scolari
Meanwhile, Argentina coach Alejandro Sabella, 59, will decide his future next week after his side lost 1-0 to Germany in the World Cup final.
Members of the squad have said they want Sabella to remain in charge for the 2015 Copa America in Chile.
"It's down to whether I feel strong enough to give 100%," said Sabella.
Brazil won the 2013 Confederations Cup, but lost in the semi-finals of the 2014 Fifa World Cup
