Spartak won on goals by Argentina striker Fernando Cavenaghi, a penalty by Brazilian Mozart, and Alexander Pavlenko. Paraguayan international Fredy Bareiro scored for Moscow Region club Saturn.
The two clubs played out a 1-1 draw at Spartak's home Luzhniki stadium last week.
Spartak, which has been going through troubled times in recent months after its captain Dmitry Alenichev was put on the transfer list and head coach resigned, will play in the final against either 2005 UEFA Cup winner CSKA Moscow or Zenit St. Petersburg.
Alexander Starkov resigned as Spartak's head coach on April 24. Last year, he took the nine-time Russian champion to second place in the Premier League and qualification for the Champions League.