The Bochum striker toe-ended in Valter Birsa's cross a minute before half-time in what proved to be the game's only goal to spark jubilant scenes at the Ljudski Vrt stadium.
Russia, who had Aleksandr Kerzhakov and Yuri Zhirkov sent off in the second half, were left to rue some poor defending and an overall disjointed performance.
Matjaz Kek kept the same eleven that started the match in Moscow, while Guus Hiddink brought in Lokomotiv Moscow's Renat Yanbaev at left-back and moved Yuri Zhirkov further forward to the left side of midfield.
The late goal in Moscow had handed Slovenia a lifeline and they began the game with real purpose, whilst the Russians, having let a commanding two-goal cushion slip, looked nervy and disjointed, particularly in defence.
The warning signs were there for Hiddink as early as the fourth minute, when a simple through-ball from Birsa inexplicably cut open the Russian defence and allowed Milivoje Novakovic a clear sight of goal, but his finish lacked confidence and was straight at keeper Igor Akinfeev.
Six minutes later there was a huge let-off for Russia when Andraz Kirm's cross from the left struck the leg of the hapless Sergei Ignashevich, who saw the ball rebound off the post with Akinfeev motionless. Dedic then beat Yanbaev to a deep cross from Novakovic, but his header lacked power and Akinfeev gathered comfortably.
As the clock ticked towards half-time, the home support erupted as Dedic got behind two Russian markers to turn in Birsa's cross and give Slovenia the lead at the break.
Hiddink made two changes at half-time, bringing on Sergei Semak and Aleksandr Kerzhakov for Pavlyuchenko and Yanbaev. But Russia's night went from bad to worse when the Dinamo Moscow striker was shown a red card by Norwegian referee Terje Hauge on 66 minutes.
Zhirkov's cross from the left was whipped in dangerously towards Kerzhakov, who got the better of his marker and slid in on goalkeeper Samir Handanovic, appearing to make a stab at the loose ball.
The challenge was interpreted as dangerous play by the referee. Closer inspection of the incident appears to show Slovenia defender Bostjan Cesar strike Kerzhakov in the face, while goalkeeper Handanovic chipped in with a retaliatory hand aimed at the striker.
Hiddink reacted by throwing on another striker, Pavel Pogrebnyak, in place of Diniyar Bilyaletdinov and the Stuttgart man almost made an immediate impact with a 12-yard header from Arshavin's cross that was held by Handanovic. As the game neared its dramatic climax, Zhirkov soared into the penalty box and hit a volley that was beaten out by Handanovic.
The ball was immediately cleared downfield to Dedic, who drilled a glorious opportunity wide, while at the other end Vasili Berezutsky came closest to finding an equaliser for Russia, volleying over from close range following Ignashevich's flick.
There were more ugly scenes in stoppage time as Zhirkov was shown a second yellow card following an altercation with a Slovenia substitute on the byeline.
Seconds from the end Novakovic had a further glorious chance to put the icing on the cake, and the final whistle brought scenes of jubilation, with Russia ultimately left to pay the heaviest of prices for not finishing the match off on Saturday.