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.
Goals by midfielders William Magrao and Maylson gave Gremio, favourites to lift the state title won for the last two years by Inter, their 13th consecutive win in the Gaucho championship.
In the Paulista, another of the many state championships in the first third of the Brazilian season, Corinthians beat Sao Paulo 4-3 to take their unbeaten run in their derby to nine matches since 2007.
A stoppage-time own goal by defender Alex Silva gave Corinthians victory after former Brazil left-back Roberto Carlos had scored one of their goals with a trademark free-kick.
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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."
Paulista leaders Santos, without Robinho who is still recovering from a minor thigh injury, beat Monte Azul 5-0 to take their tally to 69 goals in 20 matches.
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