On Monday, Russian authorities said they had opened a criminal investigation into the incident, which took place near the city center.
They did not confirm the identities, but said two local residents were injured.
Ukrainian bloggers report that as a result of the attack, Yevsyukova’s wife was also injured, who lost her leg and is in a critical condition in the hospital.
“(We) are conducting a detailed inspection of the scene of the incident and a complex of investigative measures,” the Investigative Committee of Russia said.
Footage from the scene shows a burnt-out white SUV parked on Donetsk’s main street.
The killing of Yevsyukov is seen as the latest in a series of targeted attacks by Ukraine against Russian officers, Russian-appointed military officials and pro-Kremlin public figures in the occupied territory of Ukraine and in Russia.
In November, senior Navy officer Valery Trankovsky, whom Ukraine accused of war crimes, was killed in Crimea. His murder was organized by Ukrainian intelligence, the AFP news agency reported last month, citing a source in the security services of Ukraine.
In October, Ukraine claimed responsibility for a car bomb explosion that killed an official at the Russian-occupied Zaporizhia nuclear power plant.
And in April, a government official appointed by Moscow was killed as a result of a car bomb explosion in the territory controlled by Russia in the eastern Luhansk region of Ukraine.
The city of Donetsk and part of the wider region in eastern Ukraine were first captured by Russian-backed forces in 2014, and the area has been partially controlled by Moscow since then.
For the past almost three years, some of the fiercest battles of the war have been constantly taking place in the region.