Soccer World Cup 2018 in Russia (File)
One fan was taken to hospital after cutting himself on a carriage door, while another who tried to leave the train was detained and charged with disobeying a policeman's orders.
The incident comes as England is set to play its first World Cup match in Russia, after serious violence broke out between Russian and England fans in the French city of Marseille during the Euro 2016 championship.
Regional transport police said in a statement that one of the fans "in a state of inebriation, broke the glass on the entrance door to a carriage and cut open his right hand".
He was taken to a hospital in the town of Yelets - almost 700 kilometres (430 miles) from the venue in Volgograd - but did not require inpatient treatment.
The other fan "in a state of inebriation, learning his acquaintance had been hospitalised, decided to leave the train" at the same station, transport police said.
Other England fans tried to stop him, resulting in a "conflict," and the fan "did not obey (a policeman's) lawful demands to cease acts of hooliganism," police said.
The unruly fan was taken to a police station in Yelets and charged with petty hooliganism and disobeying the orders of a member of law enforcement. Both men were also charged with drunkenness.
The charge of petty hooliganism carries a maximum term of 15 days in police cells or a fine of 2,500 rubles ($39), while being drunk in a public place is punishable by up to 15 days in police cells or a fine of 1,500 rubles.
A British embassy representative confirmed to AFP that one British man had been detained and said embassy staff were in contact with Russian authorities and were ready to offer him consular assistance.