Novak Djokovic of Serbia reacts after winning a point.Credit:
Defending Australian Open champion Djokovic was the red hot favourite against former world No.5 Anderson who was playing his first match since July after returning from knee surgery.
However,two-time grand slam finalist Anderson - now the world No.91 - showed plenty of fight before succumbing 7-6 (7-5) 7-6 (8-6) in match lasting almost two and a half hours as Serbia grabbed an unbeatable 2-0 lead.
Djokovic - aiming for a record eighth Australian Open crown this month - extended his head-to-head record against the South African to 9-2.
Serbia's Dusan Lajovic came from a set down to beat Lloyd Harris in the singles opener.
While Djokovic avoided an upset,Thiem became the ATP Cup's biggest name to fall.
Two-time French Open finalist Thiem arrived at the inaugural 24-team tournament as one of tennis'hottest properties after being runner-up at November's ATP Finals,losing to Greece's Stefanos Tsitsipas.
But world No.28 Coric pulled off a boilover,downing Thiem 7-6 (7-4) 2-6 6-3 to give Croatia an unbeatable 2-0 lead in their tie at Ken Rosewall Arena.