Spike Sheilla Castro scored 27 points for the Brazilians who twice battled back from a set down to set up a semi-final against Japan who also came through a five-setter against China.
The day's first two quarter-finals spanned nearly five hours and produced an electrifying atmosphere in the cavernous pre-World War Two venue in west London.
The United States, whose women are trying to emulate the men's side's gold in Beijing, comfortably beat Dominican Republic 3-0 despite being without injured captain and setter Lindsey Berg, and not being quite at the top of their game.
"We've got 12 people for a reason. Things happen and unfortunately it happened to me, but I can't be selfish," Berg, playing at her third and final Olympics, said.
Asked if she would be fit to return against South Korea in Thursday's semi-final, she said: "Ice, rest, massage and I'll hope for the best and if not, I'm confident Courtney (Thompson) can take my place."
South Korea came from a set down to outplay Italy and reach the semi-finals for the first time since 1976.
Brazil began the tournament slowly and group phase defeats against the United States and South Korea left them hanging on by a thread, but they have since returned to form.
Victories against China and Serbia meant they qualified for the knockout phase but they looked to have reached the end of the road against Russia, who in 2.02 metre (6ft 8ins) Ekaterina Gamova boast one of the most formidable players in the world.
However, with captain Fabiana Claudino inspiring her players and the many Brazilians in the crowd willing them on, the South Americans emerged victorious, winning a nail-biting decider 21-19 on their first match point after saving six.
"We are very happy with the result to win 3-2 in the quarter-final," coach Jose Guimaraes told reporters.
"After (the match against) Korea we tried to talk to each other about the situation of the teams going through.
"We were not playing well, we were not playing as a team, even though the preparation was really good. Since then we started to improve. Today was a fantastic match."