The three countries who won their "city finals" to progress were joined in the last four by Italy, the runners-up with the best record in the $15 million event which replaced the short-lived ATP Cup to open the 2023 season.
The US will play Poland for a place in the final while Greece meet Italy.
Tiafoe gave the Americans an unassailable 3-1 lead with his third straight victory of the season after Jessica Pegula downed Harriet Dart 6-2 6-0 in 57 minutes. Madison Keys had earlier put the U.S. ahead after rallying past Katie Swan 2-6 6-3 6-4.
The in-form Cameron Norrie had put Britain level at 1-1 with a 6-4 5-7 6-4 win over world number nine Taylor Fritz, who later teamed up with Pegula to beat Dart and Evans.
"It's a big deal," Tiafoe said. "We want to go deep...win this event so it was one of the stepping stones towards that. Today was a big one."
Poland's players and staff broke into dance after their heavyweight mixed doubles pair of Iga Swiatek and Hubert Hurkacz won the final rubber against Italy's Lorenzo Musetti and Camilla Rosatello 6-1 6-2 to wrap up a 3-2 victory.
Swiatek and Magda Linette won their singles matches to twice draw Poland level after Hurkacz and Daniel Michalski were beaten by Musetti and Matteo Berrettini to set up a thrilling finish under the roof at Pat Rafter Arena in rain-soaked Brisbane.
"Amazing," three-times Grand Slam champion Swiatek said. "It's even better winning in a team than as an individual. I'm so happy we did it together and the team effort paid off."
Greek Stefanos Tsitsipas shrugged off a time violation on his serve for discarding what he believed was a "broken" ball to outlast Borna Coric 6-0 6-7(4) 7-5 and end a three-match losing run against the Croatian.
But Greece were dragged into a deciding clash with the tie locked 2-2 and Tsitsipas and Maria Sakkari overcame Petra Martic and Borna Gojo 7-6(6) 6-4 to spark wild celebrations among fans in Perth.
The semi-finals begin in Sydney on Friday, with the final two days later.