The second edition of the international mixed team competition is a shortened version, with matches consisting of two singles matches and one doubles match.
Britain only needed the singles competition to achieve victory in Perth, as Norrie defeated Alex de Minaur for the second year in a row 6-4, 2-6, 7-6, before Katie Poulter won 6-2, 6-4. Victory over Ajla Tomljanovic.
Nouri won all three of his United Cup matches a year ago, including in straight sets against De Minaur, but struggled during the second half of the 2023 season and ended up losing nine of his last 11 matches.
The 28-year-old admitted he felt a little burned out after maintaining an intense schedule, but the off-season allowed him to reset, and his win over De Minaur was his first against a top 20 player since March.
Nouri took advantage of De Minaur's slow start, breaking serve in the first game and keeping his opponent at arm's length during the first set.
The Australian player, who is six places ahead of Nouri, ranked 12th in the world, raised his level significantly in the second set and appeared to be the better player throughout most of the deciding periods of the set.
But Norrie held on well, saving a break point in the sixth game, playing a brilliant tiebreak, seizing the initiative and double-faulting De Minaur on the first match point.
Speaking on court, Nouri said: “I had to dig deep and play very bravely in that third set. He came out at the end of the second set, he was going strong and I couldn't match his level and I had to really play to win the tie-break.
“I found good depth on my forehand in the tie-break. It's always difficult to play Alex, he's a good friend of mine. “It was a good match and a great way to start the year.”
Much of the pre-match attention was focused on De Minaur's clash with girlfriend Poulter, but the British number one ensured Australia would not have a second chance with a strong performance.
Tomljanovic was previously a top 40 player but missed most of last season after knee surgery and is currently ranked 290th.
Poulter, who is ranked 56th in the world rankings, led 4-0 in the first set, and although it was closer after that, she maintained her nerve to snatch the victory.
Britain needs to beat the defending champions, the United States, in its final group match on Sunday to ensure qualification to the quarter-finals.