Valeri Karpin's side produced a dominant first-half performance at the Arena Khimki on Saturday to make it three wins from three at the start of the new campaign.
All of Spartak's goals came in the opening 45 minutes with Jose Manuel Jurado and Pavel Yakovlev putting the away side 2-0 up within 12 minutes.
Nigerian Emmanuel Emenike fired home his second of the season and Jurado added another three minute before the half-time whistle to essentially put the game beyond doubt.
Rene Meulensteen's first game in charge of Anzhi Makhachkala ended in a 1-1 draw away to Krylya Sovetov at the Stadion Metallurg on Sunday.
After a goalless first-half, Samuel Eto'o broke the deadlock in the 59th minute.
But, with just eight minutes remaining on the clock, midfielder Roman Vorobyov's first goal of the season ensured both sides earned a share of the spoils.
In Sunday's other fixture, Terek Grozny held Rubin Kazan to a 0-0 draw, while Krasnodar earned their first win of the campaign with a 2-1 success over Amkar Perm on Monday.
Zenit St Petersburg's shaky start to the season continued on Friday as they stumbled to a 1-1 draw at home to Kuban Krasnodar at the Stadion Petrovskij.
Former Arsenal attacker Andrey Arshavin gave Luciano Spalletti's side a second-half lead, but they were unable to hold their advantage and Gheorghe Bucur levelled proceedings with 11 minutes remaining.
