Television images showed the front of a passenger bus crushed by the overpass, which was located about 3km from the Mineirao Stadium, where World Cup games are being played.
Other vehicles were also crushed underneath the wreckage, officials said.
Two people died and 19 were injured, a spokeswoman for the Minas Gerais state health department told Reuters.
A firefighters' spokesman said 13 people were rescued from the bus unscathed but the driver was killed.
A car was also crushed but it was not immediately known how many people were inside and what their condition was, firefighter spokesman Edgard Estevo da Silva told reporters.
"The vehicle is completely under" the overpass, he said, adding that the cause of the collapse was not immediately known, AFP reported.
The cause of the collapse was unclear. The overpass was part of a network of bus lanes that Brazil planned to have ready for the World Cup, but was not finished on time.
"We were traveling normally and then there was a terrible noise," Renata Soares, who said she was on the bus at the moment of the accident, told GloboNews TV.
"I am sure that more people in other cars were underneath the debris."
A World Cup semi-final match will be held in Belo Horizonte next Tuesday. The stadium has already hosted five games.
Heavy government spending on the World Cup and long delays in infrastructure projects have spurred violent street protests in Brazil over the past year, although they have died down in recent weeks.