The visiting duo completed a convincing 6-3, 7-6 (7-3), 7-5 victory to hand Kazakhstan a 2-1 lead heading into Sunday's singles.
Nishikori is the highest-ranked player Murray has ever played in Davis Cup, but he has won five of their previous six matches.
The world No. 1 will hope to get the ball rolling against Mikhail Kukushkin in the first reverse singles clash, and if he's successful, Viktor Troicki will get a shot at redemption against Nedovyesov (or Golubev) in a deciding rubber.
"They were better in crucial moments", admitted Djokovic.
Serbia's Novak Djokovic, front and his partner Nenad Zimonjic react during their Davis Cup World Group first round doubles tennis match against Kazakhstan¿s Aleksandr Nedovyesov and Andrey Golubev, in Belgrade, Serbia, Saturday, March 5, 2016. Former US Open champion Marin Cilic gave Croatia a winning start in Liege, seeing off Kimmer Coppejans 7-5, 6-3, 7-5.
Murray and Kei Nishikori both won their 12th straight match in the Davis Cup though Nishikori had it much tougher in the second singles match before finally putting away 157th-ranked Dan Evans 6-3, 7-5, 7-6 (3).
Switzerland, the champions in 2014, were without star players Roger Federer, who is injured, and Stan Wawrinka, who opted not to play in the tie in Pesaro.
Italy became the first team to reach the quarter-finals when Simone Bolelli and Andreas Seppi defeated Marco Chiudinelli and Henri Laaksonen 6-3, 6-1, 6-3 for a 3-0 win over Switzerland. But Federer and Wawrinka responded with victories to avoid the upset on the next day. "I was getting a little out of breath but I played a good match, served well, missed a few second serves but the first serve went extremely well so it was a nice start".
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France chose to play on clay outdoors on the island of Guadeloupe to neutralize Milos Raonic of Canada, but he's sidelined, as he's still recovering from the adductor he tore in the Australian Open semifinals.
The Murray brothers maintained their superb record in the competition by beating Japanese pair Yoshihito Nishioka and Yasutaka Uchiyama 6-3, 6-2, 6-4 in Birmingham on Saturday.
The winners face either Germany or the Czech Republic, champions in 2012 and 2013, in the next round. They didn't face a break point.
In Hanover, Tomas Berdych claimed a 50th Davis Cup win when he and veteran Radek Stepanek claimed a 7-6 (9/7), 7-5, 6-4 win over Germany's Philipp Kohlschreiber and Philipp Petzschner. The reverse singles on Sunday are scheduled to be John Isner vs. Bernard Tomic, and Jack Sock vs. Sam Groth.
Just as in their title triumph over Belgium in November in what was Britain's first Davis Cup triumph in 79 years, the brothers played a starring role.
Argentina lead Poland 2-0 in Gdansk as Guido Pella beat Michal Przysiezny 6-1, 6-4, 7-6 (5) and Leonardo Mayer defeated Hubert Hurkacz 6-2, 7-6 (3), 6-2.
Berdych, 7th ranked in the world, had topped Alexander Zverev to draw the Czechs level Friday.