Knapp dispatched Christina McHale 6-3, 3-6, 6-1 in the day's first match indoors at the Cleveland Public Auditorium.
Giorgi, competing in her first Fed Cup for Italy, then took advantage of 47 unforced errors by American Madison Keys en route to a 6-2, 6-1 victory.
Keys will need to rebound in Sunday's first reverse match against Knapp if the United States is to rally for its first win over the Italians since 2003.
The winner of the weekend series will go on to face the Spain-Czech Republic victor in April's semifinals.
Celebrity Families of 2014 [PHOTOS]