Playing just five days after lifting his 24th Grand Slam title overall in New York, Djokovic overcame the hot and humid conditions early on and recovered from 4-1 down in the second set to beat Alejandro Davidovich Fokina 6-3, 6-4 in Valencia.
The result secured an unassailable 2-0 lead for Serbia after Laslo Djere beat Albert Ramos-Vinolas 6-4, 6-4 in the first match of the Group C tie and secured a tie in the last eight in Malaga, which will take place in November.
"I'm on cloud nine, as you can see with everything that's has been happening lately on the tennis court," Djokovic said following his 13th straight match victory.
"Playing for Serbia, for my country, is something completely different. It's a huge responsibility and pressure but also an incredible privilege and honour, so I'm really glad that I was able to contribute with a crucial point to qualify for Malaga.
"Obviously unfortunate for the Spanish crowd, they wanted Spain to win but I think we did really well... and now we'll try against the Czech Republic tomorrow to clinch top spot in the group to hopefully get a better draw."
Djokovic said he was fully committed to helping the 2010 Davis Cup champions in their quest for a second title in the premier men's team competition.
"It was all happening really quickly within a few days and I was on a high with the US Open win and the celebration at home and leaving my family is never easy but coming to another family - my Serbian squad - gives me the greatest of joy," he added.
"At the beginning of the season, I said the Grand Slams and playing for my country were my priorities.
"I want to contribute as much as I can. Here we are, the goal is reached, we're in the last eight. Hopefully, we can keep going in the right direction."
Over in Manchester, an emotional Andy Murray gave Britain a 1-0 lead over Switzerland by outlasting Leandro Riedi 6-7(7), 6-4, 6-4 and later revealed he missed his grandmother's funeral to play in the Group B clash.
Veteran Stan Wawrinka rolled back the years to level the tie by beating Cameron Norrie 7-5, 6-4 and returned to play the doubles rubber alongside Dominic Stricker, but the Swiss were no match for Dan Evans and Neal Skupski who prevailed 6-3, 6-3.
In Group A, Italy delighted home fans in Bologna by taking a 2-0 lead over Chile as Matteo Arnaldi downed Cristian Garin 2-6, 6-4, 6-3 and Lorenzo Sonego overcame Nicolas Jarry 3-6, 7-5, 6-4.
Finland handed Croatia their second defeat in Group D with Otto Virtanen getting past Dino Prizmic 6-4, 3-6, 6-3 and Emil Ruusuvuori beating Borna Gojo 7-6(3), 6-4 in Split.
The result meant the Netherlands advanced to the knockouts.
