A distraught Troicki, who won the deciding rubber in the 2010 final, blamed himself, saying: "I probably feel the worst ever.
"I never experienced such a moment in my career, in my life. And I let my team down, and I apologise to them.
"We had chances to finish it. We didn't do it. I messed up in the crucial moments."
Djokovic added: "It hurts really badly.
"These kinds of matches happen once in maybe forever. That's it. The season is done and we're turning the next page."
Serbian captain Nenad Zimonjic was also overcome with emotion, struggling to get the words out when he fronted reporters as the players sat beside him, some burying their faces in their hands.
Troicki couldn't look up and Krajinovic was in tears as Zimonjic reflected on why the result was so devastating, paying tribute to the retiring Janko Tipsarevic as he looked back at the sad way Serbia's "golden generation" was knocked out.
Zimonjic had to stop speaking to on several occasions to regather himself as the enormity of the situation sunk in.
"The main thing is that everybody here did their part, you know, not just now, but during their career," he said. "And that's all you can say, you know. It was very emotional because it's Janko's last โฆ sorry.
"Sorry, it's not because here winning or losing, just for you to understand โฆ
"I see it as an end because it's Janko's last match."
The press pack applauded and tennis scribes were struck by the raw emotion on show.
Russia will face Canada in the semi-finals after the North Americans defeated Australia 2-1 on Friday.
Khachanov and Rublev, who have played all the rubbers between them this tournament in the absence of US Open finalist Daniil Medvedev, fell to the court in an embrace at the moment of victory after beating the Serbs.
"Both teams had a lot of chances," said Rublev. "It was like a coin toss, and we won this coin toss."
