Their wins ensured Spain and Serbia clinched their ties at the inaugural 24-nation team event, where the countries are split into six groups across Sydney, Brisbane and Perth.
The top eight from the round robin go onto the knockout phase in Sydney before one is eventually crowned the winner.
France, Argentina, Croatia and Japan also began their tournament in style, winning their ties to join day one victors Belgium, Bulgaria, Canada, Norway, Russia and Australia.
“I’m super, super happy to be here in Perth for first time in my life. It’s an amazing arena and a great crowd,” said Nadal.
“It was tough first match of the season, not an easy start. But I think I played a great game.”
Earlier, his teammate Roberto Bautista Agut crushed a hapless Aleksandre Metreveli 6-0, 6-0.
Serbian superstar Djokovic needed to produce some of his best tennis to see off the big-serving Anderson, admitting it was a testing encounter.
“What a way to start the year,” he said. “He was on fire tonight — I think the quality of tennis was quite high. “I was very pleased with the way I kind of weathered the storm.”
His teammate Dusan Lajovic outlasted Lloyd Harris 3-6, 7-6 (7/4), 6-3.
Austria’s Thiem forecast this week that 2020 would be the year that the vice-like hold of Djokovic, Nadal and Roger Federer on Grand Slam titles would come to an end.
He is tipped as one of those who could make the breakthrough, along with Alexander Zverev and Stefanos Tsitsipas. But all three have been beaten in their opening ATP Cup clashes.
Thiem, who won five titles last year, had six break points in the third set against Croatia’s Borna Coric, but failed to convert any of them.
“It’s really nice to beat him because at the end of last season I was not playing well and this will give me a lot of confidence,” said Coric, ranked 28 and who suffered six losses in a row in late 2019.
Fellow Croat Marin Cilic, a finalist at the 2018 Australian Open, used his experience to down Austria’s Dennis Novak 6-7 (7/4), 6-4, 6-4, and ensure Croatia won the tie in Sydney.
The most dominant win of the day came courtesy of Japan, who are without the injured Kei Nishikori but crushed Uruguay 3-0 in Perth, dropping just 15 games across their two singles and one doubles rubbers.
Yoshihito Nishioka routed higher-ranked Pablo Cuevas 6-0, 6-1 and veteran Go Soeda thrashed Martin Cuevas, Pablo’s younger brother, 6-1, 6-3 before a tighter doubles encounter.
French duo Gael Monfils and Benoit Paire, both in the top 25, led their country to a 2-1 Group A win over Chile, while Argentina scraped past Poland by the same score.
But it was a struggle for the South Americans with Polish number one Hubert Hurkacz upsetting world number 14 Diego Schwartzman in three sets.
Guido Pella won the other singles before Maximo Gonzalez and Andres Molteni teamed to down Hurkacz and two-time Grand Slam winner Lukasz Kubot in the deciding doubles.
Some US$15 million in prize money is at stake in the tournament, with up to 750 singles and 250 doubles ATP rankings points up for grabs.
Under the innovative tournament rules, on-court coaching is allowed and team zones are in the corners of the court, rather than on the sidelines by the umpire.