CSKA Moscow midfielder Alan Dzagoev got the decisive goal in a result that moved Russia up to 20 points to retain the summit in Group B, leaving them requiring just a draw from their remaining qualifier against minnows Andorra.
Hosts Slovakia, who desperately needed to win to keep their chances of qualifying alive for the EURO finals in Ukraine and Poland, started in a lively manner spurred on by a capacity 11,000 home crowd under pelting rain.
In the 23rd minute Russia created the first serious chance through their skipper Andrei Arshavin, who dribbled into the area surrounded by three Slovak defenders. However, his close-range attempt was confidently denied by Mucha.
The hosts replied positively and also created their own scoring opportunity in the 37th minute forcing Russian keeper Vyacheslav Malafeyev to pull off a diving save to clear Marek Hamsik's pass into the box onto unmarked Miroslav Stoch.
Two minutes after the interval Filip Holosko also had a chance to put the hosts into the lead, but Malafeyev performed another remarkable save to deny the Besiktas striker's powerful shot from the edge of the box.
Russia replied with a 20-metre effort by Arshavin, but Mucha also produced a save to parry the attempt. In the 65th minute Roman Shirokov missed a perfect scoring chance, sending the ball wide with an unopposed 10-metre shot from Dzagoev's pulled-back pass from the goal line.
But in the 70th minute Dzagoev finally had Russian supporters on their feet, firing the ball into the net past a helpless Mucha with a 36-metre shot.