Direct coaches from Moscow to Bratislava ply along with the train Moscow - Chop with departure at 22:13 from Kievskaya station in Moscow to Bratislava Main station in Bratislava through Kosice, Zilina, Bratislava-Vinogradi station.
Please note that when going to Slovakia there are "cost-effective prices" when the ticket price doesn't change because of departure station in Russia (without seat reservation), and also gives you a discount when buying more than 1 ticket and if buying round-trip tickets. The cheapest way is to buy round-trip tickets for 5 people.