Serbia finished second behind Czech Republic, but clinched their place in the Final Eight with a win over Spain.
Novak Djokovic beat Alejandro Davidovich Fokina just five days after winning the US Open to help Serbia to a 3-0 win over Spain in Valencia.
Whoever comes out on top between Serbia and Britain will then face the winner of Italy v Netherlands in the semi-final.
Reigning Davis Cup champions Canada will continue their title defence against Finland, who came through as runners-up in Group D after surprise wins over Croatia and the United States, while Canada topped their group ahead of Italy.
The winner of Canada v Finland will take on either Czech Republic or Australia in the semi-final. Australia finished runners-up to Britain.
The Final Eight takes place in Malaga, Spain, and begins on November 21.
Winner Group A v Runner-up Group D
Canada v Finland
Winner Group C v Runner-up Group B
Czech Republic v Australia
Runner-up Group A v Winner Group D
Italy v Netherlands
Runner-up Group C v Winner Group B
Serbia v Britain