Nadal beat Pablo Cuevas 6-2, 6-1 to secure Spain's win in the best-of-three encounter against Uruguay in Perth ahead of the doubles.
Behind Novak Djokovic, Serbia joined Australia in the ATP Cup final eight.
He had beaten Monfils in all 15 previous meetings and that flawless record never looked in danger as he dominated the Frenchman from the outset. No matter how many grand slams he wins, he will never be the greatest for me.
Kyrgios has beaten Djokovic both times they've played and said although he expected the 32-year-old to overtake Roger Federer in terms of most major singles titles won, it doesn't mean he would claim the mantle as the greatest male player ever. "For me personally - I don't care right now, I've come this far - I feel like he just wants to be liked so much that I just can't stand him". "I had a lot of them", Cilic said.
As the Frenchman began to show the effects of the humidity, Djokovic stepped up the pressure and eased to victory in 90 minutes in front of hundreds of chanting Serbian fans. Another break of Georgian Nikoloz Basilashvili's serve was enough for him to win the opening set.
Out of anger at losing the second set, despite having had two chances to close the match, Paire smashed two of his racquets, threw several water bottles and needed to be reassured by his teammates. The Group A match will now be decided by doubles.
Japan also edged Georgia 2-1 to also go to 2-0.
The 16-time Grand Slam victor saved nine break points to get past Gael Monfils 6-3, 6-2, before stepping back on the court with partner Viktor Troicki to take on Nicolas Mahut and Edouard Roger-Vasselin in the doubles.
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.
Roberto Bautista Agut was equally ruthless in getting Spain started when he beat unranked 19-year-old Franco Roncadelli 6-1, 6-2.
"It was phenomenal. Serbian people, I mean, I didn't expect, to be honest, so many of them staying all the way through and basically being on a Christmas Day with us post-midnight, which is fantastic", Djokovic said. "I played a very solid match, my serve worked better than the first day", Nadal said.
South Africa's Kevin Anderson shares Novak Djokovic's concern about potential air quality at the Melbourne Open after the Australian bushfires.
Cilic, who has a career-high ranking of No 3 but slipped to 38 after an injury-shortened 2019 season, has now won both his singles matches in Sydney at the new global men's team event.
Other winners Monday included Japan, Croatia, South Africa and Austria.
The result also extended Djokovic's record over Monfils to an incredible 16-0, a head-to-head series dating back nearly 15 years.