Djokovic pulled Serbia level in front of empty stands in Innsbruck with victory over Jan-Lennard Struff after Dominik Koepfer had defeated Filip Krajinovic but Djokovic and Nikola Cacic were then beaten in a deciding doubles.
They did well to battle back from a set down against Kevin Krawietz and Tim Puetz and led by a break in the final set, but the German duo levelled and Djokovic threw his racket in anger after Serbia failed to break at 5-5.
The rubber went to a deciding tie-break and it was Krawietz and Puetz who pulled off a 7-6 (5) 3-6 7-6 (5) victory.
Virtually the whole Serbian team was in tears after a tough loss to Russia in the quarter-finals two years ago, with Viktor Troicki holding himself responsible.
Troicki is now captain, and he looked emotional again as he said: “What can I say after this match? Let’s say the bad luck is following us in Davis Cup for the last couple of years in doubles unfortunately. I hope it’s going to turn. I hope we’ll still have a chance to compete this year in the Davis Cup.”
Djokovic did not turn up to the press conference.
Germany will be heavy favourites to beat Austria on Sunday and set up a quarter-final against the winners of Group C, which looks likely to be Great Britain, who were 2-1 victors over France.
Italy became the first team to book their spot in the last eight, Lorenzo Sonego and Jannik Sinner winning their singles rubbers against Colombia to follow up victory over the USA.
The Russian Tennis Federation, who are the favourites to lift the trophy, began their campaign with a 3-0 victory over Ecuador and they will face defending champions Spain on Sunday for a place in the last eight.
Kazakhstan put themselves in pole position in Group B with a 2-1 victory over Sweden while Australia defeated Hungary but seem certain to be heading home.