Guus Hiddink's side, rated by many as dark horses for the tournament as a whole, led 2-1 from the first leg but were undone by Zlatko Dedic's first-half opener.
It was enough to secure a priceless ticket to South Africa for Matjaz Kek's side on the away goals rule.
Russia, who finished with nine men, fell behind when Dedic converted a Valter Birsa cross towards the end of the first half.
Substitute Alexander Kerzhakov was dismissed on 66 minutes after a clash with keeper Samir Handanovic and Chelsea's Yuri Zhirkov also saw red in the dying seconds.
Portugal sealed their place in the finals by repeating their 1-0 first-leg win against Bosnia-Herzegovina.
The tricky away tie was successfully negotiated without Cristiano Ronaldo as Jose Raul Meireles struck the only goal of the game 10 minutes into the second half following good play by Nani and Simao Sabrosa.
Sejad Salihovic was sent off for dissent 13 minutes from time as Bosnia failed to breach the Portugal defence.
Dimitris Salpigidis was the hero as Greece won 1-0 in Ukraine.
The only goal of the two legs was scored just after the half-hour mark as he latched on to a Giorgos Samaras pass to cleverly beat Andriy Pyatov.
Alexandros Tzorvas was in fine form to keep Alexei Mikhailichenko's team at bay in the second half and earn the Greeks their first place in the finals since 1994.
Anatoliy Tymoshchuk wasted a late opening as Ukraine bowed out in front of their home fans.
Earlier in the day, Algeria put paid to Egypt's hopes of playing in South Africa with a 1-0 victory in Sudan.
After conceding a late, late goal to the Egyptians at the weekend, Algeria gained their revenge with Antar Yahia's volleyed strike after 40 minutes.
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