In an entertaining match at London's O2 Arena, both teams exerted consistent pressure on their opponents' serve until Bryan and Sock buckled in the seventh game.
World number one Djokovic comfortably saw off the South African fourth seed Kevin Anderson.
The Swiss maestro lost to Alexander Zverev in the last four at the ATP Finals, which brought his 2018 campaign to a premature halt.
German Zverev is the youngest finalist at the event since Juan Martin del Potro in 2009.
The Serb will take on Kevin Anderson in the semifinals.
Top seed Djokovic is already through to the semi-finals after beating Alexander Zverev in the Guga Kuerten Group on Wednesday.
Coming into his final Lleyton Hewitt Group match, Thiem knew anything other than a straight-sets win would end his ambitions of reaching the semi-finals at London's O2 Arena.
It will offer $15 million (13.24 million euros) in prize money and up to 750 ATP Rankings points to the winners.
Tsitsipas started the season ranked 91st in the world but after reaching the finals in Barcelona and Toronto, losing both to Rafael Nadal, he is now up to 15th.
Djokovic is chasing a sixth ATP Finals title and his first in the prestigious season-ending event at London's O2 Arena since 2015.
French veteran Julien Benneteau had said the 20-Grand Slam champion was able to use his reputation to secure favourable scheduling.
Anderson won the first 11 games and needed only 64 minutes to demolish the woeful Nishikori in a remarkably one-sided group stage clash at London's O2 Arena.
Gasquet's decision, made on Monday when he cited an unspecified injury, leaves France relying on Jo-Wilfried Tsonga and Lucas Pouille for the singles rubbers against Croatia in Lille.
The Serb is a red-hot favourite to draw level on six titles at the season finale with Roger Federer.
Australian Open finalist Cilic came into the match with the dispiriting record of five consecutive defeats against the young German but broke Zverev at the first opportunity.