All four goals came in the second half as the hosts threw off the shackles of a frustrating opening 45 minutes, Dzyuba opening the scoring with a volley just before the hour.
Ozdoev rifled home the pick of the bunch three minutes later and Dzyuba took his tally for the campaign to eight before Aleksandr Golovin compounded Scotland's misery in Moscow.
The result puts Russia on the brink of qualification, while Steve Clarke's Scotland have the significant consolation of knowing they are already assured of a play-off spot courtesy of their Nations League exploits.
Clarke made six changes from the 4-0 loss to Belgium, handing debuts to John Fleck and Michael Devlin, but Russia controlled proceedings from the outset.
FULL TIME | Russia 4 - 0 Scotland— Scotland National Team (@ScotlandNT) October 10, 2019
It is all over in Moscow. Goals for Dzyuba, Ozdoev and Golovin give the home side win. pic.twitter.com/GV3MPhSNAM
Despite that, they struggled for clear-cut openings before the break, Fedor Kudryashov firing off target when well placed at the back post to underline their lack of a cutting edge.
Stanislav Cherchesov's side, who won the reverse fixture 2-1, continued to press forward and Aleksey Ionov saw a strike blocked by Charlie Mulgrew, with goalkeeper David Marshall remaining untested in the opening 30 minutes.
He was relieved to see Golovin's curling free-kick dip narrowly over the crossbar as Russia's frustration grew further in a scoreless first half.
Dzyuba eventually broke the deadlock when he connected with Golovin's corner and Ozdoev lit up the contest with an unstoppable drive from outside the box.
There was further punishment to come for Scotland, who have now shipped 13 goals across four consecutive losses, with Dzyuba poking into the net amid some questionable defending.
Golovin added further gloss with a coolly taken side-footed effort to cap a fine performance from Russia.
What does it mean? Russia on the brink, Scotland eye alternative route
Russia are eight points ahead of third-placed Cyprus with three games remaining, making their qualification all but certain. On this display, they will fancy their chances of getting the job done in Nicosia at the weekend.
For Scotland, the remaining trio of group matches represent an opportunity to build some form heading into the play-offs.
Danger man Dzyuba at the double
Cherchesov's side were on top throughout a one-sided contest but the breakthrough seemed like it may never come.
That was until talisman Dzyuba stepped up and got his side off the mark, breaking down Scotland's resolve and leaving them vulnerable to the quick one-two punch that followed. He added a third, doubling his own tally, for good measure and is in contention to be the top scorer in qualifying.
Woeful outing for sorry Scotland
This was a chastening outing for Scotland, who can in truth point to very few positives, if any.
That they have a play-off spot in the bag should not excuse the apparent lack of attacking conviction on display in the Russian capital.
Key Opta facts
- Russia remain unbeaten against Scotland (P4 W2 D2 L0), with all four matches coming in Euro qualifiers, winning each of the last two.
- Scotland have lost four consecutive matches in all competitions for the first time since 2004, under Berti Vogts.
- Russia have won 10 of their last 11 qualifying matches, losing only against Belgium back in March.
- Only Israel's Eran Zahavi (10) has scored more goals in EURO 2020 qualifying so far than Artem Dzyuba (8).
- Aleksandr Golovin has been directly involved in three goals (1 goal, 2 assists) in a game for Russia for the first time since their World Cup 2018 opener against Saudi Arabia in June 2018 (1 goal, 2 assists).
Scotland host group whipping boys San Marino on Sunday, while Russia head to Cyprus on the same day.