Swiatek needed just 52 minutes to dispatch Prisacariu 6-0 6-0 to extend her singles winning streak to 19 in her second competitive match since becoming world No.1 for the first time.
"It took us a few years, step by step, to progress and qualify finally," the 20-year-old Swiatek said in an on-court interview.
"Hopefully in the finals we're going to show even more progress. I'm really proud of us."
Swiatek's victory made it 3-0 for Poland in the best-of-five qualifier for the finals in November.
The 2020 French Open champion beat Mihaela Buzarnescu 6-1, 6-0 on Friday.
Italy and Kazakhstan also advanced on Saturday.
Camila Giorgi beat Harmony Tan of France 6-2 6-0, giving the Italians their third straight win.
Elena Rybakina made it three in a row for Kazakhstan over Germany by beating Angelique Kerber 4-6 6-3, 7-5.
The last time Germany missed the Billie Jean King Cup finals was in 2013, when it was still called the Fed Cup.
Belgium are already in the finals after receiving a walkover with opponents Belarus being suspended from international team competitions following Russia's invasion of Ukraine.
In Prague, Marketa Vondrousova routed US Open champion Emma Raducanu 6-1 6-1 to give the Czech Republic a 2-1 edge over Britain.
Raducanu, slowed by a blister on her foot, had debuted in the competition with a victory over Tereza Martincova on Friday but struggled against Vondrousova on a cold morning at the outdoor clay court.
Vondrousova, a former French Open finalist playing on her home court, needed just 62 minutes to beat the 19-year-old Raducanu, who took a medical timeout after the first set.
The Czechs have won the Cup six times since 2011 though they are without their three highest ranked singles players — No.2 Barbora Krejcikova, No.7 Karolina Pliskova, No.28 Petra Kvitova — and No.1-ranked doubles player Katerina Siniakova.
Canada, led by US Open runner-up Leylah Fernandez, Spain and the US are all in good positions to advance to the finals with 2-0 leads ahead of Saturday's matches.
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