The hard-fought victory in Zilina leaves the Dutchman and his side needing just a single point from their final group game to secure their place at next year's tournament
A solitary 70th minute strike from Alan Dzagoev proved enough to secure an impressive three points for Dick Advocaat's side while ending Slovakia's qualification hopes.
The visitors asked the questions in the opening stages as their central defensive pairing of Vasili Berezutski and Sergei Ignashevich patiently passed the ball around at the back before feeding the men who could do the damage further forward.
One such move eventually led to an eighth minute chance for Roman Pavlyuchenko, who fired in a powerful drive from 25 yards which was well held by Slovakia goalkeeper Jan Mucha.
There was a feeling of deja vu seconds later as the Tottenham man unleashed an almost identical effort from a similar position. The Slovakia keeper was equal to this effort as well, diving to his right to block this time.
Andrey Arshavin, who also plies his trade in north London, then took up the mantle as he peppered Mucha’s goal with efforts.
He came closest to breaking the deadlock in the 23rd minute when he burst through the heart of the Slovak defence with a piercing run, leaving him one-on-one with Mucha. The stopper had read the danger, however, and raced from his line to make a crucial block.
For their part, the hosts grew in stature as the game progressed, and this was in no small part due to the fact that star player Marek Hamsik was growing in influence.
The Napoli man’s probing passes were causing the Russia rearguard particular problems and in the 35th minute he sent Miroslav Stoch in on goal with just the goalkeeper to beat. But, like Mucha at the opposite end, Vyacheslav Malafeev darted out to thwart the winger.
The second half was almost indistinguishable from the first 45, in that neither side overexerted themselves in their efforts to open the scoring.
Filip Holosko and Arshavin swapped long-range efforts from the edge of the box, neither of which ever really looked destined to breach the respective goalkeeper’s net, before Russia passed up a glorious chance in the 65th minute.
Dzagoev wriggled away from Erik Jendrisek on the left byline and rolled a perfect cutback towards Roman Shirokov, who was unmarked 12 yards from goal, but the Zenit St Petersburg man somehow managed to blaze wide.
But after watching his team-mate blow the opportunity he had crafted so superbly, Dzagoev decided to go it alone five minutes later with spectacular results. The CSKA Moscow midfielder picked the ball up 30 yards from goal and unleashed a ferocious drive which deflected off Ignashevich and looped up over Mucha and into the net to give Russia a precious 1-0 victory.
The final round of fixtures will be played simultaneously on Tuesday and see Russia host Andorra while Slovakia travel to meet Macedonia.