Gremio beat Esportivo 2-0 and Inter lost to Caxias by the same score in the Gaucho championship in the southern Brazilian state of Rio Grande do Sul.
Uruguayan Fossati said his future hung on a good result at home to Cerro from his home country in their Libertadores Cup Group Four match on Wednesday.
"I confirm that if I don't manage to transmit calm (to the team), I have to let another (coach) come here," Fossati told reporters after the Caxias match, for which he rested several first team players.
Both teams had a player sent of just before half time when Corinthians striker Dentinho and Sao Paulo's Washington clashed.
"Everyone fights to the death against us," said Roberto Carlos, who joined Corinthians this season as they seek to win the Libertadores for the first time in their centenary year.
"We won with sweat but it was a very important victory. I'm confident, I feel good in this shirt."
