Gauff, a two-time Grand Slam champion, beat Sakkari 6-3, 6-2 in an hour and 26 minutes to put the US in front.
The 21-year-old had lost in three sets to world number 42 Jessica Bouzas Maneiro when the US took on Spain in Group A on Monday, but she shot out of the blocks against Sakkari and quickly racked up a 5-0 lead.
Sakkari showed some resistance, winning three straight games, but Gauff held serve to take the opening set before breaking the Greek twice in the second set to wrap up the win.
Greece's Stefanos Tsitsipas then earned a 6-4, 7-5 win over world number nine Taylor Fritz to level the tie.
The 27-year-old was afforded little time to savour his first top 10 win since 2024, however, returning to the court just half an hour later alongside Sakkari to take on Gauff and Christian Harrison.
Momentum swung wildly in the hard-fought decider, with both teams taking a set apiece to send the match into a tiebreak.
The US raced into a 5-0 lead in the tiebreak, and though the Greek duo rallied to level at 6-6 a perfectly placed volley from Harrison squeezed through Tsitsipas's legs and sealed a 4-6, 6-4, 10-8 victory.
"I thought it was great level throughout, we just played free and aggressive," Gauff said.
Switzerland advance to semi-finals
Switzerland beat Argentina 2-1 as Belinda Bencic fired the European side into the semi-finals by first winning her singles match before returning for the decisive doubles match.
Bencic first beat Solana Sierra 6-2, 6-2 to give Switzerland the lead but Sebastian Baez levelled the tie when he defeated Swiss veteran Stan Wawrinka 7-5, 6-4.
Bencic then teamed up with Jakub Paul to beat Maria Lourdes Carle and Guido Andreozzi 6-3, 6-3 to book Switzerland's spot in the last-four. Bencic remains unbeaten in the tournament with a 6-0 win-loss record this year.
"It's a great feeling to start a season well," Bencic said. "It gives you a lot of confidence that you come out of the blocks pretty good, and you already have so much play, of course, before Australian Open is always very important."
Poland face Australia while Belgium take on the Czech Republic in the other quarter-final ties this week.
The 18-nation event in Perth and Sydney serves as a warm-up for this month's Australian Open - the season's first Grand Slam.
