Captain Dzyuba opened the scoring 12 minutes after the break in Moscow when he crashed home a close range volley from a corner and put Russia three goals up with a tap-in 20 minutes from the end.
Magomed Ozdoev scored the goal of the night with a thumping longe-range effort seconds after Dzyuba had got the ball rolling for the hosts, while the lively Aleksandr Golovin completed the rout six minutes from time to leave the Scots pinning their Euros hopes on the Nations League playoffs.
The comfortable win puts Russia, second in Group I, three points behind already-qualified Belgium with three matches left and eight ahead of third-placed Cyprus, who snatched a late 2-1 win at Kazakhstan earlier and who host Russia on Sunday.
A draw would do in Nicosia thanks to their 1-0 win over the Cypriots in Kaliningrad last month, and the better head-to-head record would be enough to guarantee Stanislav Cherchesov's side at least second place and a spot in the multi-host nation event.
Despite being eliminated from the main qualifying phase following their fourth straight defeat, Scotland could still make it to the Euros through the UEFA Nations League play-offs.
Steve Clarke's side won Group C1, securing both a place in League B for the next edition of the Nations League and a chance to book a place in the Europe-wide tournament.
As it stands they will take on Group C2 winners Finland in March's semi-finals and one of Norway or Serbia in the final, although all three are still in contention for the tournament through the main qualification route.
The Finns are currently second to Italy in Group J in and have a good chance of making it, while Serbia are a point behind second-placed Portugal in Group B and the Norwegians two behind Scandinavian rivals Sweden in Group F.
Should any of those teams qualify, the next best placed teams in the League C groups would take their place.