Victory at the Shenzhen Bay Sports Centre Arena helped the Italians become the first team to reach three successive finals in the elite women's team competition since the Czech Republic won three straight titles from 2014-16.
Paolini and Errani secured the tie by beating the Ukrainian pair of Marta Kostyuk and Lyudmyla Kichenok 6-2, 6-3 to draw loud applause from the sizeable crowd of Italian fans who remained in their seats until close to midnight in southeastern China.
"Today the crowd was amazing. We had the support of all these guys, the Italian fans. We enjoyed a lot playing in this competition," said Errani, who joined the fans in chanting the traditional rallying cry of "Jia You" - literally "add oil".
"It was a tough day for Jasmine. She's doing an incredible job. It's not easy to play doubles after a tough singles match but she's so strong.
"I'm really happy to have her as my doubles partner."
Former world number three Svitolina earlier squandered the chance to hand her country an unassailable 2-0 lead in the tie and the opportunity to fight for their maiden trophy on Sunday after she went down 3-6, 6-4, 6-4 to an inspired Paolini.
On the ropes
The 31-year-old broke Paolini in the third game of their contest with some aggressive hitting from the baseline and took the opening set with a sizzling backhand winner to leave five-times champions Italy on the ropes.
Paolini responded by breaking early in the second set but Svitolina took control from there and was a point away from a 5-2 advantage, before losing her way again with a flurry of uncharacteristic errors to surrender the set.
Having levelled the clash, Paolini fought fiercely to hold in an eight-deuce game early in the decider and went ahead 4-2 before Svitolina clawed he way back, but the determined Italian would not be denied as she raised her level again to prevail.
The tie-levelling victory came after Ukraine's Kostyuk took apart Elisabetta Cocciaretto 6-2, 6-3, leaving world number eight Paolini needing a first victory over Svitolina in their third meeting to keep her team alive.
Paolini, who won Olympic gold partnering Errani in Paris last year and the French Open doubles title in June, said that she was confident heading into the deciding contest despite playing for two hours and 25 minutes.
"We've been playing doubles together for the last two years, more or less every tournament," Paolini said.
"It's good to have doubles as well in the Billie Jean King Cup. We enjoy playing doubles a lot and we enjoy playing in this competition. She has won this competition four times. She loves the tournament.
"I have to thank the team for supporting us ... and also the fans. They were amazing.
"We just want to enjoy the final on Sunday. It's amazing to play this competition. The expectation that I have is that all the fans come back and cheer for us."
Up next for Italy is either the US, who will seek a record-extending 19th title in the competition previously called the Fed Cup, or four-time runners-up Britain, with those two teams playing in the other semi-final on Saturday.