Ukraine took a 1-0 lead in the 68th minute when Milan's Andriy Shevchenko was brought down in the area, earning his team a penalty which Serhiy Rebrov calmly fired home.
A 1-0 victory would have sent Ukraine through instead of Slovenia, on the away-goals rule. But the Slovenians equalised in the 77th minute when Zlatko Zahovic's free-kick ricocheted off Ukraine's defensive wall straight to Miran Pavlin, who sent scored past goalkeeper Oleksandr Shovkovskyi.
Ukraine had a limited choice of players following the suspension of Dmytro Parfenov and Andriy Husin after the first leg, and the first half of this second leg brought no goals. The Slovenes dominated, exerting tight control on an extremely slippery, snow-covered field, organising speedy attacking moves and easily repelling Ukraine's rather disoriented sorties.
Slovenian zeal was somewhat broken, though, when Zahovic was shown with a yellow card after being brought down inside Ukraine's box, and put his hands to his head in disbelief.
One of the rare dangerous moments for Slovenia was a cross from the right by Oleh Luzhnyi which was saved by goalkeeper Mladen Dabanovic.
Ukraine's luck changed in the second half, with incessant pressure on Dabanovic's goal.
Ukraine were left with 10 players in the dramatic last minutes after an injured Shevchenko left the field. Shovkovskyi abandoned his goal to assist the Ukrainians' desperate attacks, even managing a shot on goal.
In Kozani, Greece, last night, a goal by Thomas Kyrparissis gave Greece a 1-0 victory over Bulgaria in a friendly international. The match was part of both teams' preparations for the preliminary rounds of the 2002 World Cup.
The score was kept low by excellent performances by both goalkeepers. Bulgaria's Zdravko Zdravkov and his Greek counterpart Andonis Nikopolidis each made a series of decisive saves.
In addition, Stanimir Stoilov and Stilyan Petrov both cleared off the line in the 17th and 45th minutes respectively.
Greece launched one attack after the other in the second half, but the Bulgarian resistance held out. In the 61st minute, the Ajax forward Nikos Machlas passed to Kyparissis, whose ground-hugging right-footed shot from the edge of the box was saved by by Zdravkov.
Four minutes before the final whistle, Fanis Katergiannakis, who replaced Nikopolidis in the goal, dived to save a shot from point-blank range by Ivailo Petkov.
Ukraine: Shovkovskyi, Luzhnyi, Fyodorov, Holovko, Vashchuk, Dmytrulin, Kandaurov (Kovalyov, h-t), Skachenko (Moroz, 58), Kossovskyi (Popov, 74), Rebrov, Shevchenko.
Slovenia: Dabanovic, Karic (Osterc, 74), Rudonja, Milanic, Galic, Milinovic, Novak, Ceh, Udovic (Acimovic, 57), Pavlin, Zahovic.