World number two Djokovic shrugged off his recent elbow injury to give Serbia a winning start in their quarter-final in Belgrade with a 6-3, 6-4, 6-2 victory over 24th-ranked Albert Ramos Vinolas.
Viktor Troicki, the world 39, then made it 2-0 by seeing off 19th-ranked Pablo Carreno Busta 6-3, 6-4, 6-3.
"I am satisfied with my game, especially with the serve," said Djokovic after claiming a 12th successive Davis Cup singles win.
"It worked well today and I won a lot of points off my first serve."
Djokovic, who skipped the Miami Masters because of an elbow injury and went into the tie in Belgrade with just seven wins on tour in 2017, dropped just one point on serve against Ramos Vinolas.
Five-time winners Spain are without Nadal after the 14-time Grand Slam champion opted to stay at home to prepare for the clay-court season.
Team captain Nenad Zimonjic and Troicki will face Marc Lopez and Jaume Munar in Saturday's doubles where a win will give Serbia a semi-final date against either France or Britain.
On a perfect day for all four of the quarter-final hosts, Australia were 2-0 up on the United States in Brisbane, closing in on a semi-final date against Belgium who were 2-0 ahead of Italy in Charleroi.
In Rouen, Lucas Pouille and Jeremy Chardy moved nine-time champions France to the verge of a fifth semi-final in eight years with straight sets wins against a British team missing Andy Murray who sat out the tie with an elbow injury.
Pouille gave France a winning start with a 7-5, 7-6 (8/6), 6-3 victory over Kyle Edmund before Chardy then eased past Dan Evans, playing on clay for the first time in three years, 6-2, 6-3, 6-3.
Chardy, only France's ninth best singles player, was competing in his first Davis Cup singles rubber in six years.
Pouille, the world number 17 and his team's top singles player in the absence of Jo-Wilfried Tsonga and Gael Monfils, recovered from 2/5 down in the second-set tie-breaker before dominating the deciding set.
"I have lost matches in my career before, I'm going to lose more," said Edmund.
France can wrap up the tie on Saturday when Nicolas Mahut and Julien Benneteau face Jamie Murray and Dominic Inglot in the doubles.
"I am always nervous, but I am a lot more relaxed now than I was yesterday," said France skipper Yannick Noah, who has won the Davis Cup both as a player and captain.
Nick Kyrgios and Jordan Thompson gave Australia an unexpected 2-0 lead against the United States in the countries' 47th meeting in the competition.
World number 79 Thompson stunned world number 15 Jack Sock 6-3, 3-6, 7-6 (7/4), 6-4 before Kyrgios saw off John Isner 7-5, 7-6 (7/5), 7-6 (7/5).
The controversial Kyrgios pushed Roger Federer to the brink in the Miami semi-finals last week and he carried that form into Friday's match.
"I'm trying to knuckle down and compete for every point and that's something I haven't done in the past," said Kyrgios.
"I'm still very emotional out there. I'll still challenge the umpire, I'll still say what I think, I'll still throw my racquet, I'll still swear, but I'll compete every point."
In Charleroi, Steve Darcis beat Paolo Lorenzi 6-7 (3/7), 6-1, 6-1, 7-6 (7/4) while David Goffin eased past Andreas Seppi 6-4, 6-3, 6-3 as Belgium led Italy, who knocked out defending champions Argentina in the first round.
