It remains unclear what caused the 62-year-old diplomat to die.
The Sudanese foreign affairs minister, who confirmed the information in a statement, did not give a cause of the death.
He is the fourth Russian ambassador to die on the job over the past year, including United Nations envoy Vitaly Churkin and India envoy Alexander Kadakin.
"He was found dead in the swimming pool of his residence at 7:05 p.m.", Omar al-Mokhtar, a police spokesman, told AFP.
In a statement carried by Sudan's state-run SUNA news agency on Thursday, the Sudanese foreign ministry confirmed Shirinsky died in Khartoum on Wednesday and it offered its "warm condolences" to the government and people of Russian Federation.
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"Doctors were summoned but were unable to save him", Konyashin said on state-run Rossia-24 television.
The press secretary at Russia's embassy, Sergei Konyashin, said there were signs the ambassador had suffered a heart attack.
Embassy staff called an ambulance but "Shirinsky could not be saved", Zakharova said. Russian ambassador to Turkey, Andrey Karlov and senior diplomat Petr Polshikov were both shot dead on the same day on 20 December in Ankara and Moscow respectively.
Sudan's foreign ministry also praised the Russian ambassador's efforts to enhance diplomatic relations between the two countries while in office, but provided no details on the circumstances surrounding Shirinsky's sudden death.
Shirinsky served in the Egyptian, Yemeni, Saudi and Rwandan embassies before his posting to Sudan, according to Russia's Sputnik news agency.