The world number one had his Italian counterpart on the ropes in the second singles rubber at 5-4 up, but allowed Sinner to recover and win 6-2, 2-6, 7-5 to force the doubles decider.
Djokovic and Miomir Kecmanovic, who had won the opening singles rubber against Lorenzo Musetti, were defeated by Sinner and Lorenzo Sonego 6-3, 6-4 and leave Malaga disappointed.
"For me personally it's a huge disappointment, because I take the responsibility, obviously having three match points, being so close to win it," Djokovic told reporters.
"Yeah, it's unfortunate really. This is sport. When you lose for your country, you know, the bitter feeling is even greater."
Djokovic said it was "tough to swallow" the defeat, in a competition he was desperate to win with Serbia for the second time after their only triumph in 2010, which he was part of.
The 36-year-old Serb, a record 24-time men's Grand Slam winner, said he thought world number four Sinner's level might dip, but the Italian was relentless.
"I know Jannik's quality, but, you know, I was thinking maybe he's gonna drop a little bit, you know, in the doubles, but he didn't," said Djokovic.
"He kept the level extremely high, and at the end of the singles match, also third set, I had a lot of chances to break him and was in his service games, but every time he needed he found the serve."
Djokovic congratulated Sinner, who has now beaten him three times in four matches across 11 days.
Sinner won in the ATP Finals group stage last week but Djokovic responded by beating the Italian to win the trophy.
"They deserved it, they played really well, particularly Jannik, in singles against me and then doubles, as well," added Djokovic.
"He barely missed the ball the entire match ... you can only say congrats."
