The new mixed team event is being played across three cities in Australia, with Britain, captained by Tim Henman, taking on the hosts and Spain in Sydney.
With each match taking place across two days, the 2023 season had to begin before the New Year, but Britain made a sharp start and lead Australia 2-0.
Norrie, who spent most of the off-season playing exhibition matches across the world, impressed in a 6-3 6-3 victory over Alex De Minaur, who is ranked 10 places below him at 24th.
The British number one said afterwards: “I’m feeling great. Especially first match of the year, there’s a lot of nerves. No tougher task than to play Demon in Australia. It was not easy.
“I ran a lot, I absolutely drenched three shirts. I really enjoyed the atmosphere and nice to get the win.”
The 23-year-old will be looking to break into the top 100 this year after a career disrupted by injury so far and was too consistent for the erratic Zoe Hives, fighting back from 3-0 down in the second set to win 6-4 6-3.
Swan, ranked 145, said on BT Sport: “It’s the first match of the season. I haven’t played for about two and a half months so I was pretty nervous coming out but what a way to start the year. Such a cool event.
“Cam played a great match and it was almost more nerve-racking watching him. It took a lot of pressure off knowing we were 1-0 up. It’s a great start for the team and looking forward to supporting the guys playing tomorrow.”
Dart will hope to repeat her Billie Jean King Cup win from last month against Ajla Tomljanovic in the first match on Friday, while Dan Evans will take on Jason Kubler followed by a mixed doubles clash.
The USA are also 2-0 up against the Czech Republic in the other match in Sydney while Greece lead Bulgaria 2-0 in Perth following a tense battle between Stefanos Tsitsipas and Grigor Dimitrov, and Italy and Brazil are locked at 1-1 in Brisbane.