Russia's Marat Safin, No. 2 in the men's game, secured his first singles win of the new season when he beat Jan-Michael Gambill 6-4, 6-7 (6-8), 6-3.
However, it was not enough to keep Russia in the hunt for the title as a Monica Seles-inspired U.S. team moved to the brink of a place in Saturday's final.
Seles' 6-3, 6-3 win over Elena Likhovtseva and Gambill and Seles' 4-6, 6-2, 6-2 triumph in the decisive mixed doubles gave the U.S. a 2-1 aggregate win in group B.
GULF IN CLASS
Hingis has worked diligently on her fitness and serve and volley game in the close season, and the benefits were evident as she thrashed 56-ranked Nicole Pratt 6-1, 6-1 in 48 minutes.
Federer, 19, gave Switzerland an unassailable 2-0 lead in the group A match by beating Richard Fromberg 6-3, 6-2 after squandering three match points in a row.
The 3-0 score was completed when Federer and Hingis beat Fromberg and Pratt 6-1, 6-3 in the mixed doubles.
Hingis, 20, said: "I have improved constantly and I think there is more to come.
"I have been working hard in the holidays, and I think I can get even better. But no-one can ever say they are perfect."
Australia, weakened by the absence of injured Mark Philippoussis, cannot now reach Saturday's final.
Switzerland, with two victories, are close to achieving that, although South Africa can still top group A.
South Africa, the defending champion, meet Thailand on Thursday and then plays Switzerland on Friday.
REVIVED INTENSITY
Safin, who was sluggish in defeat to Belgium's Olivier Rochus on Monday, showed revived intensity in beating Gambill on his fourth match point. But the 2-1 defeat left Russia bottom of group B.
The U.S. meets Belgium and the Slovak Republic face Russia in the final group matches on Thursday.
Belgium, who had to qualify via a play-off win over Japan, can still make the final, but the pairing of Rochus and Kim Clijsters need to beat the U.S. by the aggregate score of 3-0.
After striking 17 aces, Safin admitted Gambill was "unlucky" to lose the final set, but there were signs that his power serve and all-round mastery of the game was returning.
"I am determined to keep improving like this, because I will have a good chance of success at the Australian Open."
Gambill, who saved three match points, begged to differ after relinquishing a 3-2 final set lead.
"I have to be blunt - I don't think he won the match, I think I played worse at the crucial times. It is very disappointing," the 23-year-old said.
Seles remains on course to be the first player to win the Cup twice - and with different nations. In 1991, she took the title for her native Yugoslavia with Goran Prpic.
The 27-year-old said: "I am not playing very well at the moment. I am sluggish and will have to improve."
The Australian Open begins in Melbourne on January 15.