Holders Czech Republic showed how tenaciously they will fight to retain the Davis Cup as Tomas Berdych and Lukas Rosol won a record-breaking doubles match against Swiss duo Marco Chiudinelli and Stanislas Wawrinka.
Their seven-hour and two-minute victory included a marathon 24-22 final set and put the Czechs 2-1 ahead in the tie in Geneva and they need to win only one of the two remaining singles rubbers.
The winners play either Kazakhstan or Austria in April's quarter-finals - Austria kept their tie against Kazakhstan alive as Julian Knowle and Alexander Peya won the doubles to leave the hosts leading 2-1 in Astana.
Serbia and France cruised into the quarter-finals, where the French will face Argentina, who beat Germany with a day to spare, by taking a 3-0 lead.
France, who are bidding for a 10th Davis Cup title, are also unbeatable in their tie against Israel.
Depleted top seeds Spain are still alive in Canada later but trail 2-1.
Serbia have an unassailable 3-0 lead against Belgium and will face the United States if they complete victory over Brazil, having forged a 2-1 lead despite doubles specialists Bob and Mike Bryan suffering a stunning loss in the doubles.
Marcelo Melo and Bruno Soares claimed a famous 7-6 6-7 6-4 3-6 6-3 win to keep Brazil alive and condemn the American brothers to only the third defeat of their Davis Cup partnership.
Elsewhere, Italy lead Croatia 2-1.