The new case, reported on Tuesday night, comes after the Goa Police claimed last week that the arrest of 50-year-old Francis Pereira would end the series of desecrations in the state.
In the latest incident, nearly a dozen crosses were damaged at the cemetery near Fatima Chapel in Madkai, located 35 km from Panaji.
Responding to comments by opposition MLAs during zero hour, Chief Minister Parrikar said: "Police have got definite clues. In three or four days they would solve the case and arrest the culprits."
Leader of Opposition Chandrakant Kavlekar raised the issue of mass desecration of graves. "People are worried because the latest incident has happened while Francis Pereira who was arrested recently for allegedly single-handedly destroying religious structures, is in police custody," Mr Kavlekar, a Congress MLA, said.
Mr Kavlekar said the latest desecration incident proved that there was more than one person involved in the serial desecration incidents in Goa, in connection with which Pereira was arrested last week.
Mr Parrikar also told the House that Pereira during his interrogation had confessed to desecrating 146 religious structures across Goa.