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South Africa now lead the tie 2-1 going into Sunday's reverse singles matches, with Van der Merwe up first against Blaz Kavcic. Later Ruan Roelofse will take on Grega Zemlja if required.
Despite losing the first set in a tie break after the locals were 5-2 up and serving for the set, Klaasen and Van der Merwe came back to win the second set in a tiebreaker.
Slovenia's doubles team of Kavcic and Zemlja were up 4-3 in the tie break, but South Africa rallied to win the next two points on the visitors serve, and held their nerve to serve out the set.
The third set saw the Slovenian resistance crumble as South broke both the second and fourth games before they ran out to claim the set 6-1.
In the fourth set South Africa broke again in the fifth game as they knocked the confidence of the Slovenians. Van der Merwe served out the match in the 10th game to win the set 6-4.
"They played really well. There was one sloppy game when Izak served at 5-2 [in the first set], they kind of got a little bit more confidence, and we let it slip there," said team captian John-Laffnie de Jager speaking after the match.
"We knew they were not beating us, and I just kept on telling my guys be positive and keep creating opportunities, and those guys will start making mistakes.
"We were hoping to be up going into Sunday, giving Izak a lead for his match. It was a big match today," said De Jager.
Klaasen, who played a particularly solid game said his team had to hang in there when the contest became tight.
"The whole match we were ahead, and we kept having to reinforce that we were doing the right thing," said Klaasen.
"It's tough out there when things aren't going exactly the way you think it should be."
Heading into the final day Van der Merwe is positive after his doubles victory with Klaasen and about ending the tie in his rubber with a win.
"For me it's a confidence boost to win the doubles, I saw him [Kavcic] serving so that helps," said Van der Merwe.
"I'm looking forward to tomorrow(Sunday), I think he's going to look at it as a clean slate and I might take a little bit of confidence into that match."
Davis Cup results around the world:
World Group quarter-finals results:
At Castellon (clay - outdoor)
Spain 2 Austria 1
Friday
Nicolas Almagro (ESP) bt Jurgen Melzer (AUT) 6-2, 6-2, 6-4
David Ferrer (ESP) bt Andreas Haider-Maurer (AUT) 6-1, 6-3, 6-1
Saturday
Oliver Marach/Alexander Peya (AUT) bt Marcel Granollers/Marc Lopez (ESP) 3-6, 6-4, 6-4, 7-6 (14/12)
Sunday (10:00 GMT)
David Ferrer (ESP) v Jurgen Melzer (AUT)
Nicolas Almagro (ESP) v Andreas Haider-Maurer (AUT)
At Monte Carlo (clay - outdoor)
France 1 USA 2
Friday
Jo-Wilfried Tsonga (FRA) bt Ryan Harrison (USA) 7-5, 6-2, 2-6, 6-2
John Isner (USA) bt Gilles Simon (FRA) 6-3, 6-2, 7-5
Saturday
Bob Bryan/Mike Bryan (USA) bt Julien Benneteau/Michael Llodra (FRA) 6-4, 6-4, 7-6 (7/4)
Sunday (09:00 GMT)
Jo-Wilfried Tsonga (FRA) v John Isner (USA)
Gilles Simon (FRA) v Ryan Harrison (USA)
At Prague
Czech Republic 2 Serbia 1
Friday
Tomas Berdych (CZE) bt Viktor Troicki (SRB) 6-2, 6-1, 6-2
Janko Tipsarevic (SRB) bt Radek Stepanek (CZE) 5-7, 6-4, 6-4, 4-6, 9-7
Saturday
Radek Stepanek/Tomas Berdych (CZE) bt Nenad Zimonjic/Ilija Bozoljac (SRB) 6-4, 6-2, 7-6 (7/4)
Sunday (from 11:00 GMT)
Tomas Berdych (CZE) v Janko Tipsarevic (SRB)
Radek Stepanek (CZE) v Viktor Troicki (SRB)
At Buenos Aires
Friday
Argentina 2 Croatia 1
Marin Cilic (CRO) bt David Nalbandian (ARG) 5-7, 6-4, 4-6, 7-6 (7/2), 6-3
Juan Martin del Potro (ARG) bt Ivo Karlovic (CRO) 6-2, 7-6 (9/7), 6-1
Saturday
David Nalbandian/Eduardo Schwank (ARG) bt Marin Cilic/Ivo Karlovic (CRO) 3-6, 7-6 (8/6), 6-3, 6-7 (6/8), 8-6
Sunday (13:30GMT)
Juan Martin del Potro (ARG) v Marin Cilic (CRO)
David Nalbandian (ARG) v Ivo Karlovic (CRO)
