Paolini, who like Trevisan also won her singles against France, then dispatched Anna-Lena Friedsam 6-3 6-2.
It is the first time Italy have reached the semi-finals of the competition since 2014.
Captain Alicia Molik changed her line-up and world number 155 Storm Hunter rewarded her faith by posting a 7-6(2) 6-4 victory against Anna Danilina, ranked a lowly 802nd.
Later on Thursday, reigning champions Switzerland face the United States in Group A needing to win to stay in the hunt.
Canada can secure their place in the semi-finals if they beat a Poland side without world number one Iga Swiatek.
Persons:
Billie Jean King, Jasmine Paolini, Germany's Anna, Lena Friedsam REUTERS, Marcelo Del Pozo, Martina Trevisan, Eva Lys, Trevisan, Anna, Lena Friedsam, Alicia Molik, Storm Hunter, Anna Danilina, Yulia Putintseva, Kim Birrell, Hunter, Ellen Perez, Martyn Herman, Toby Davis
Organizations:
La, Germany, France, Slovenia, Kazakhstan, Switzerland, United States, Thomson
Locations:
Seville, Spain, Italy, Kazakhstan, France, Germany, Australia, Slovenia, United, Canada, Poland