In the reprisal attacks, several missiles struck the city of Kharkiv with two children among those injured, the Ukrainian military said.
Tensions between the United States and Russia have reached new heights amid the ongoing Ukraine conflict, with a series of developments marking a dangerous escalation.
Officials said that Moscow has dispatched a panel of inquiry to probe into the incident.
This attack came after Moscow launched a major aerial assault on Ukraine.
The attack happened Saturday during a training session at the Western Military District, according to TASS, which cited the Russian defense ministry.
The Russian defence ministry on Wednesday said its armed forces are ready to take all necessary measures for the safe evacuation of Indian citizens from the eastern Ukrainian city of Kharkiv.
Mizintsev said that places for temporary accommodation and rest have been set up at the checkpoints.
Russia's Defence Ministry on Saturday said that they are pulling back forces from two areas in Ukraine's Kharkiv region where a Ukrainian counteroffensive has made significant advances in the past week.
Ukraine's intelligence chief says Russia is sending more troops into Kherson -- the largest city in Ukraine under Russian occupation -- and may be preparing to defend it against a Ukrainian counteroffensive, despite previously saying that Russian forces were leaving the city.
Russia has informed the United Nations Security Council that Russian buses are ready at crossing points to go to the eastern Ukrainian cities of Kharkiv and Sumy to evacuate Indian students and other foreign nationals who are stranded there, amidst the raging conflict in the East European country.
"Our primary focus is to get Indian students out of conflict zones in eastern Ukraine," the MEA spokesperson said.
'With profound sorrow we confirm that an Indian student lost his life in shelling in Kharkiv this morning,' the MEA tweeted.
Foreign Secretary Harsh Vardhan Shringla said all Indian nationals left Kyiv on Tuesday after the Indian embassy issued an advisory, asking them to get out of the Ukrainian capital urgently.
India's comments came hours after Russia claimed that some Indian students in Kharkiv were taken hostage by Ukrainian security forces.