Knapp posted a hard-earned 6-3 3-6 6-1 win against Christina McHale and was followed on court by Camila Giorgi, who overcame a rankings gulf of 47 places to thrash 37th-ranked Madison Keys 6-2 6-1.
It meant the US would need a major turnaround of fortune at the Public Auditorium venue on Sunday, in the three remaining rubbers consisting of the reverse singles and doubles, to earn a semi-final place.
Australia built a 2-0 lead over Russia, who picked a significantly under-strength team for the tie in Hobart.
Casey Dellacqua beat 18-year-old Irina Khromacheva 6-0 6-2, and Samantha Stosur overcame 16-year Veronika Kudermetova 6-4 6-0.
There are 20 Russian women ranked higher than Khromacheva, who sits at 230th on the WTA Tour list, while Kudermetova is down at 650th.
Germany earned a precious 2-0 lead over Slovakia in Bratislava, with Andrea Petkovic digging deep and saving a match point in the second set on her way to a 2-6 7-6 (9/7) 6-2 victory over Australian Open runner-up Dominika Cibulkova in the opening rubber.
Angelique Kerber followed that by beating Daniela Hantuchova 7-6 (11/9) 6-1 as Germany seized command of the tie.
Rain in Seville meant Spain and the Czech Republic could only get through one rubber on the first day, with the home side nudging 1-0 ahead thanks to Carla Suarez Navarro's 6-1 6-4 win over Barbora Zahlavova Strycova.