The 37-year-old, playing his first competitive tennis since losing in the third round of the US Open, and his 21-year-old team-mate won 6-3, 3-6, 6-3 against Aristotelis Thanos and Petros Tsitsipas, the younger brother of Stefanos, who had to withdraw.
On Saturday, Miomir Kecmanovic had given Serbia their first point by beating Thanos 6-3, 6-3 before Djokovic blew away Ioannis Xilas, ranked 770 in the world, 6-0, 6-1 in just 45 minutes.
Djokovic said on Saturday that it "may be the last time" he plays in front of a Belgrade Davis Cup audience because of a change in format.
"Some young tennis players who will defend the colours of the national team need to take over the helm," he said.
