The home team, who also needed the points to remain in the fight for the league title, went ahead twice, in the eighth minute through Diego Souza and again five minutes into the second half through Helinho.
Fernando Uribe pegged them back immediately after the opener and then Rodinei got the Rio club a deserved equaliser eight minutes before the final whistle.
The result leaves Flamengo languishing six points behind Palmeiras with six games remaining.
Sao Paulo are in fourth, nine points off the lead.
Internacional, who were playing Atletico Paranaense later on Sunday, are two points behind Flamengo in third.
(Reporting by Andrew Downie; editing by Clare Fallon)
