Zlatko Dedic scored the only goal just before the break when he stabbed in a Valter Birsa cross as Slovenia overcame a 2-1 deficit from the first leg in Moscow to secure a ticket to South Africa on away goals after a 2-2 aggregate draw.
Victory for the Slovenians, ranked 49th in the world, takes them through to their second appearance in the finals after making their debut in 2002.
"These are dreams, dreams, but we all earned this," Dedic told Slovenian television.
"(The boys) became immortal," Slovenia coach Matjaz Kek added. "On the playground they showed that the heart is worth more than millions."
It was a bitter end for Russia's Dutch coach Hiddink, who failed to guide his team to a major tournament for the first time in his career.
Hiddink led three teams, his native Netherlands (1998), South Korea (2002) and Australia (2006) to the World Cup and also steered Russia to Euro 2008.
RUSSIA OUTPLAYED
Wednesday's match was one of Russia's worst under Hiddink as they were outplayed and outmuscled by the hard-working Slovenians, with Russia President Dmitry Medvedev and Chelsea's billionaire owner Roman Abramovich watching from the VIP box.
The Russians rarely threatened Slovenia's goal in the first half with strike pair Andrei Arshavin and Roman Pavlyuchenko held in check.
"Our wish, our goals, our dreams came true. This is a happy ending from Hollywood. The boys deserve respect, they left their heart out for Slovenia," Kek told a news conference.
