Bilyaletdinov, 27, joined Spartak Moscow in January following a disappointing spell with Everton but has struggled to regain his top form.
Zenit St Petersburg winger Bystrov, who like Bilyaletdinov played at Euro 2008 where Russia surprisingly reached the semi-finals under Advocaat's predecessor Guus Hiddink, was also left out of the squad announced on Friday.
In another surprise move, Dutchman Advocaat recalled Sporting midfielder Marat Izmailov, who had not played for the national team since 2006.
Izmailov, once regarded as one of the most talented Russian players and a veteran of the 2002 World Cup and Euro 2004, has struggled to overcome numerous injuries in recent years.
But the 29-year-old winger has shown glimpses of his old brilliance this season while playing for his Portuguese club.
Advocaat, who will step down as Russia boss following the Euros to take charge of Dutch club PSV Eindhoven, also called up two newcomers, Lokomotiv Moscow midfielder Magomed Ozdoyev and Dynamo Moscow defender Vladimir Granat.
Euro 2012, co-hosted by Poland and Ukraine, starts on June 8 with Russia playing in Group A along with Poland, 2004 European champions Greece and the Czech Republic.