Filip Krajinovic beat world No.345 Viktor Durasovic 6-2 7-5 to give Serbia a 1-0 advantage on Ken Rosewall Arena, while on Qudos Bank Arena Kamil Majchrzak overcame some second-set stutters to beat Michail Pervolarakis 6-1 6-4.
Majchrzak's win leaves countryman Hubert Hurkacz, ranked No.9 in the world, with the chance to seal the tie if he can beat world No.4 Stefanos Tsitsipas in the match of the day.
World No.8 Casper Ruud can take Norway into a deciding doubles rubber if he beats Serbia's world No.33 Dusan Lajovic, who is filling the void left by Novak Djokovic.
Earlier, Argentina pair Diego Schwartzman and Federico Delbonis barely raised a sweat while Spain cruised despite the absence of Rafael Nadal.
The Argentinian pair both won their matches 6-1 6-2 against Georgia's Nikoloz Basilashvili and Aleksandre Metreveli respectively at Qudos Bank Arena, Schwartzman particularly impressive in his disposal of the world No.22.
Their doubles pair then secured a clean sweep, also winning 6-1 6-2.
On Ken Rosewall Arena, Spain's Pablo Carreno Busta fought from breaks down in both sets to defeat plucky Chilean Alejandro Tabilo 6-4 7-6 (7-4).
Ranked 119 spots higher than Tabilo, Carreno Busta also came from 3-1 down in the tiebreak to secure a tense straight-sets victory.
Roberto Batista Agut was more clinical, trouncing Chile's world No.17 Cristian Garin 6-0 6-3 to render the doubles a dead rubber.
The pair showed they would still be a force in the teams event without Nadal, who has arrived in Melbourne ahead of the Australian Open after a preparation disrupted by a positive COVID-19 case.
"It was very early on the first of the year to play tennis, but I went to bed very early," world No.19 Batista Agut said.
"Today was a good start for me, a very solid game, really happy. I like these conditions, I like this court and Sydney's a great city."
The third edition of the $10 million ATP Cup has 16 teams divided into four groups playing at two venues at Sydney's Olympic Park.
The absence of Djokovic, Nadal and Roger Federer for various reasons has robbed the event of some star power, but 14 of the world's top 20 players are in action as they prepare for the start of the Australian Open on January 17.
Russia, headed by world No.2 Daniil Medvedev, are defending champions and will begin their tournament against France on Sunday.
Both those countries are in the same group as hosts Australia, who face the stacked Italy outfit on Sunday night.
