Russians Andreeva and Shnaider, who were competing as individual neutral athletes, took the opening set but had to settle for silver after the Italian pair fought back.
"(It's) not easy after the first set, just wanted to do better in the second set. It's so important... we gave everything and we got the gold medal," Paolini said.
"Obviously we were struggling in the first set, in the end, we made it great," Errani added.
Spain's Cristina Bucsa and Sara Sorribes Tormo beat Czech pair Linda Noskova and Karolina Muchova to win the bronze medal.
Errani, the former doubles world number one, became the seventh player to complete the career Golden Slam in women's doubles having won all four majors and the Olympic title.
Errani also became the oldest player to win an Olympic tennis gold, snatching the record that Novak Djokovic, one month her junior, had claimed two hours before by winning the men's singles final.
It was Italy's first-ever gold medal in an Olympic tennis event, a day after compatriot Lorenzo Musetti defeated Felix Auger-Aliassime to win bronze in men's singles.
Errani and Paolini said they had to settle their nerves to come back from a disappointing first set.
"During the first set we had to slowly find our way into the game," Errani told reporters.
"It was also physically challenging. It was a lot of nervous tension, a lot of pressure," added Errani, who had physio for a muscle issue after the seventh game.
The Italian pair, however, quickly got a foothold in the contest, serving better and hitting deeper as they took the second set, before coming out on top in the super tiebreak.
"The second set was a complete change. We went for it at the beginning," Errani said. "From that moment on also Jasmine has loosened up and we put more pressure on Andreeva and Shnaider and we found the right approach.
"We also tried more smashes. We didn't play close to the net. We got into the game, got into the fight and I can say that both I and Jasmine moved a lot better and that did it."
Andreeva, 17, became the second youngest player in history to win a medal in tennis at the Olympics, after Jennifer Capriati won singles gold in Barcelona 32 years ago at the age of 16 years and 132 days.
The Russian pair were happy with silver on their Olympic debut.
“We are proud of it. We could do better but first time, we are happy with the result," Shnaider said.
