David Marshall had rarely been troubled before three goals in 13 minutes around the hour mark made it five defeats in seven qualifiers for the visitors.
Artem Dzyuba netted following two corners – the second game running Scotland have conceded twice in the same circumstances – and Magomed Ozdoev thumped home from long range.
Clarke handed debuts to Michael Devlin, John Fleck and Lawrence Shankland, the latter from the bench, but he is still left seeking a way to turn around Scotland's form after losing 13 goals in four successive defeats.
The latest loss officially ended Scotland's remote hopes of a top-two finish and left them in fifth place in Group I, four points behind third-placed Cyprus.
With six centre-half options unavailable, Devlin came in alongside Charlie Mulgrew despite being dropped by Aberdeen on Saturday following a 5-0 defeat by Rangers. Clarke sprung a surprise at right-back with Liam Palmer preferred to Stephen O'Donnell for his second cap, the first coming in the 3-0 defeat in Kazakhstan which opened the campaign.
Fleck started in the absence of the suspended Scott McTominay and injured Kenny McLean, who withdrew on Wednesday, while Oliver Burke was handed the striker's role.
Russia were making their first appearance in Moscow since knocking Spain out of the World Cup and they were eager to impress more than 70,000 fans.
The home full-backs pushed right up and Scotland survived some dangerous balls into the box with the help of some decent blocks and clearances.
Golovin curled a free-kick which brushed the top of the net after being tripped by Fleck and the Monaco playmaker forced the first save in the 40th minute when Marshall parried his 20-yard strike to safety. Soon after, Callum McGregor made a crucial penalty-box block.
Scotland had their first-half moments without ever getting a shot away. Burke caused problems up to a point with his running with the ball, but his decision-making and execution let him down at the crucial moments.
Robert Snodgrass almost got into good positions from the Alaves player's lay-offs but lacked the pace to trouble the hosts.
Shankland came on at half-time and showed some good touches as Scotland improved in possession in the opening stages.
But they conceded from Russia's first attack of the half in the 57th minute. Golovin swung over a corner and Dzyuba held off Mulgrew and volleyed into the top corner from eight yards.
The hosts doubled the lead three minutes later after stretching Scotland. The ball was laid back to Ozdoev 25 yards out and the midfielder lashed his shot into the top corner.
The chances kept coming. Dzyuba chipped a shot on to the bar from a tight angle three minutes later after Scotland appeared to have averted a counter-attack and the skipper dragged a shot wide after losing Devlin from a throw-in
The third came in the 70th minute after another corner. Russia took it short this time but Scotland switched off before Golovin swung a cross beyond the back post. Mario Fernandes sneaked round the back and headed back into the six-yard box and the outnumbered Dzyuba ultimately stabbed the ball home.
Substitute Ryan Christie almost had a golden chance to pull one back after a slack pass from Russia goalkeeper Guilherme but Georgi Dzhikiya produced a goal-saving slide tackle.
Christie soon got Scotland's first shot on target but hit it straight at the keeper from 25 yards.
That was as good as it got for the Scots. Dzyuba soon set up Golovin to steer a shot just inside the post and the final whistle came as a relief to the visiting fans.