World number one Djokovic needed four hours and 57 minutes yesterday to defeat Mikhail Kukushkin 6-7 (6/8) 7-6 (7/3) 4-6 6-3 6-2 to pull Serbia level at 2-2 with Kazakhstan in Belgrade.
Viktor Troicki then beat world number 200 Aleksandr Nedovyesov 6-2 6-3 6-4 to put 2010 champions Serbia into a July 15-17 quarter-final with defending champions Great Britain.
Djokovic hit 19 aces and 37 winners but committed 94 unforced errors in his epic win over Kukushkin.
World number two Murray needed four hours 55 minutes in Birmingham to defeat Kei Nishikori 7-5 7-6 (8/6) 3-6 4-6, 6-3 in another epic duel as his side took an unbeatable 3-1 lead over Japan.
Murray had seemed set for a comfortable afternoon against sixth-ranked Nishikori by taking the first two sets against a player he had defeated five times in six previous encounters.
But Nishikori battled and was even a break to the good in the deciding set before Murray stormed back for victory.
"There were long rallies, he was pushing me quite far off the baseline. I was trying to keep the points short but in the fifth I had to grit my teeth, fight hard and I managed to get the win."
Murray hit 10 aces and 40 winners but suffered 97 unforced errors in the gruelling encounter.
In Melbourne, record 32-time winners United States
wrapped up a 3-1 win over Australia but the result was overshadowed by a bitter row between fiery Aussie teammates Bernard Tomic and Nick Kyrgios.
- Aussies in dispute -
======================
Tomic challenged the commitment of Kyrgios to the Davis Cup after his team-mate's withdrawal from the Kooyong tie with illness and back trouble.
Yet there was controversy during the match when Tomic was overheard during a changeover questioning Kyrgios's absence from the tie.
While in conversation with team captain Lleyton Hewitt, television microphones caught Tomic muttering: "Nick's sitting down in Canberra. Bullshit he's sick."
Tomic was also heard saying "that's twice he's done it", in possible reference to Kyrgios's withdrawal from the tie against the Czech Republic last year.
Croatia, the 2005 champions, needed a final rubber to defeat last year's runners-up Belgium in Liege when 19-year-old Borna Coric swept past Kimmer Coppejans 7-6 (7/5) 6-2 6-2.
Italy, the 1976 champions, had already made sure of their last-eight place on Saturday against 2014 winners Switzerland, playing without Roger Federer and Stan Wawrinka, in Pesaro.
They completed a 5-0 rout yesterday and will face Argentina for a semi-final spot.
Argentina, runners-up three times since 2006, beat World Group debutants Poland in Gdansk when Leonardo Mayer defeated Michal Przysiezny 6-7 (4/7) 7-6 (7/4) 6-2 6-3 in the first of the reverse singles.
The remaining quarter-final will see France, who swamped Canada 5-0, take on 2012 and 2013 champions the Czech Republic, who defeated Germany 3-2 in Hanover.
Earlier, Philipp Kohlschreiber had levelled the tie after Tomas Berdych retired injured after losing the first two sets 6-3 7-5.
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