They haven't won the women's nations tournament since 2000.
Garrison had to make do this week without Serena Williams, who is nursing a hand injury suffered at Wimbledon.
Russia, who have won two of the last three FedCup crowns, were also missing some big names, with reigning US Open champion Maria Sharapova and Svetlana Kuznetsova sidelined by injuries.
Taiwan lost 3-2 to Croatia in their Fed Cup tie in Split on Sunday.
The day started disastrously for the visitors as Ana Vrljic thrashed Chan Chin-wei 6-0, 6-1. That left Hsieh Su-wei needing to win her match against Nika Ozegovic to keep Taiwanese hopes alive.
Things were looking desperate when she lost the first set 6-1 but Hsieh battled back to win the second set 7-5. In the deciding third set Ozegovic reasserted the dominance she had displayed in the opening set to win 6-1 and take the match and the tie for Croatia.
Chan Yung-jan and Chuang Chia-jung claimed a consolation win in the doubles beating Jelena Kostanic Tosic and Vrljic 7-6, 6-3.
RSS