Tsitsipas shook off signs of rust and showed his fighting qualities before sealing the match with an ace.
Despina Papamichail earlier put Greece 1-0 up with a 3-6 6-4 6-1 win over Isabella Shinikova, and Maria Sakkari can seal the tie with victory over Viktoriya Tomova when Group A action resumes on Friday.
The $15 million tournament - which replaces the short-lived ATP Cup men's team event as the 2023 season opener and concludes with the final in Sydney on January 8th - features 18 countries with ties played in Brisbane, Perth and Sydney.
Earlier, Taylor Fritz and Madison Keys began their build-up for next month's Australian Open with solid displays to give the United States a 2-0 lead over the Czech Republic in Sydney.
Fritz began with a 6-3 6-4 win over Jiri Lehecka before Keys downed Marie Bouzkova 6-4 6-3 to put the Americans in control of the Group C tie.
"In a team event, first match of the year, there are always some nerves, so it's great to come out and get the team ahead," said Fritz, who saved both break points he faced in the match.
The USA can expect a tougher challenge on day two as world number three Jessica Pegula takes on double Wimbledon champion Petra Kvitova. Frances Tiafoe will then face Tomas Machac.
A mixed doubles match would decide the tie if it was locked at 2-2.
In Brisbane, Italy and Brazil were tied at 1-1 in Group E after Lorenzo Musetti downed Felipe Meligeni Alves 6-3 6-4. Beatriz Haddad Maia earlier put the South Americans ahead with a 6-2 6-0 win over Martina Trevisan.
Britain made a convincing start as Cameron Norrie eased past Australia's Alex de Minaur 6-3 6-3 before Katie Swan doubled their advantage with a 6-4 6-3 win over Zoe Hives.
"It is a tough task to play 'Demon' in Australia on this court, him playing at home. He loves playing here," Norrie said.
France took a 2-0 lead over Argentina after Alize Cornet and Arthur Rinderknech won their matches against Maria Carle and Francisco Cerundolo.
Switzerland have the upper hand in their tie against Kazakhstan thanks to wins by Belinda Bencic and Marc-Andrea HueslerMarc-Andrea Huesler.
