Some commentators have called for Brazil to appoint a foreign coach and attempt to hire former Barcelona manager Pep Guardiola.
A new coach will be chosen in early January.
Menezes had been in charge since replacing Dunga shortly after the 2010 World Cup.
His sacking was not a major surprise, but the timing was.
It came two days after Brazil won the annual Superclasico de las Americas by beating Argentina in Buenos Aires and less than a week after CBF president Jose Maria Marin said he would decide on Menezes's fate in the new year.
In 40 games, Menezes's side won 27, lost seven and drew six. But with Brazil qualifying as hosts for the 2014 World Cup, most of the games were friendlies, often against weak opponents.
Yesterday's decision was not unanimous. "I don't think we should be changing tack at this time," Sanchez said. "I was overruled."
Among the favourites to take over are Luiz Felipe Scolari, who guided Brazil to World Cup triumph in 2002, and Tite, who took Corinthians to the Copa Libertadores title earlier this year. Brazil's next match is against England at Wembley on February 6.