Beaten 2-1 by Slovakia in Lille, Russia must now win its final game against Wales on Monday and hope other results go its way to ensure qualification for the knockout phase.
Outside the pitch, tensions persisted.
French police arrested 36 football fans -- some of them from England -- in the center of Lille, France, on Wednesday evening during clashes between fans and police, police officials said. Sixteen fans were sent to a hospital, police said.
After the match, a brief spasm of street violence broke out and police used tear gas to calm the crowd.
Authorities in France expelled four people for public disorder arrests and 16 people were arrested stemming from recent incidents in Marseille and on a France-bound train.
The match was a disappointment for Russia. Set to host the World Cup in two years time, Russia was outplayed by a Slovakia side inspired by two fabulous goals, courtesy of Vladimir Weiss and Marek Hamsik.
Hamsik's strike in particular was one to remember, a beautiful curling shot from the edge of the penalty area which flew past the despairing dive of Russia goalkeeper Igor Akinfeev.
Napoli midfielder Hamsik also set up Weiss' goal with a superb pass.
Denis Glushakov's 80th-minute header gave Russia faint hope but it failed to find the equalizer it so badly craved.
Russia and Slovakia will now turn their attention to their other Group B rivals England and Wales, who play in Lens -- about a 30-minute drive from Lille -- on Thursday.
Miserable tournament for Russia
This defeat caps a miserable Euro 2016 campaign so far for Leonid Slutsky's side, which claimed a draw in its opening match against England in a game which was marred by crowd trouble.
Russia was handed a suspended ban from Euro 2016 and told it could be kicked out of the tournament if its fans persisted in causing trouble.
European football's governing body, UEFA, took action after scenes of disturbance marred Russia's opening game in Marseille last Saturday.
Russia was fined €150,000 ($168,000).
Dozens of fans were injured in clashes outside the Stade Velodrome in the French port city, while Russia fans were charged with "crowd disturbances, use of fireworks and racist behavior"
inside the stadium.
Burst of conflict after the Russia-Slovakia match
Russian Foreign Minister Sergey Lavrov says some Russian fans have behaved "unacceptably" but said that it is impossible to ignore "outrageous provocations" by English football fans.
The Russian Foreign Ministry said it summoned the French ambassador to Moscow over the detention of Russian football fans and notified the diplomat about "discriminatory measures" taken against some fans.
The ministry specifically referred to an incident Tuesday where Russian fans on a bus were detained as they were heading from Marseille to Lille for the Russia-Slovakia match.
The Nord-Pas-de-Calais-Picardie Prefecture, where Lille is located, announced punishments for people involved in public disturbances.
Three Russians and one Ukrainian are being expelled from France for public disorder offenses, the prefecture said.
Sixteen people were arrested in the region on Wednesday.
Six of them are Russian citizens involved in the violence in Marseille last Saturday, authorities said. Five people traveling to France on a train from London were arrested for public drunkenness. Their train was stopped at a station in northern France before reaching Lille.