In Saturday's championship match, the 21st seed will go up against the winner of the other semi-final between second seed Maria Sharapova of Russia and Petra Kvitova of the Czech Republic.
If she triumphs on Saturday she will be only the second Italian woman to win the French Open, two years after Francesca Schiavone's breakthrough triumph and the first player ranked outside the world top 20 to win in Paris since 1976.
"I have no words, it's incredible," she said.
"It was very windy and one side was easy and the other side more diffcult. It's an amazing feeling. I did not expect this and I can't believe it."
US Open champion Stosur was bidding to become the first Australian woman to win the French Open title since Margaret Court in 1973.
Two years ago she was the first Australian to reach the final in Paris since Wendy Turnbull in 1979, but on that occasion she lost in straight sets to Schiavone.
It was a similar task that faced her on Thursday, pitting her weight of shot and physical presence against Italian guile and court-craft in the shape of the pint-sized Errani.
She had won their five previous encounters.
After an 80-minute rain delay, the players got the first of the day's semis underway in bright sunshine and it was the 28-year-old Australian who settled in fastest.
A comfortable service hold to open was followed by a first break of serve with her trademark top-spin forehand again doing most of the damage.
Errani knew she had to vary her shot-making to try and unsettle her stronger opponent and she did so with great effect in the next game, drawing a rash of errors from the sixth seed which ended in a break of serve to love.
The Italian, winner of three claycourt titles in the lead-in to Paris, promptly held to love to level at 2-2 in what had quickly turned into a clash of contrasting styles.
Serves were held comfortably until the 11th game when Stosur again lost her touch and timing, netting an easy volley at the net on the third break point to Errani. The 21st-seeded Italian served out to take the first set.
Stosur struggled to hold serve at the start of the second set as Errani continued to run her ragged at times with her angled and varied shot-making, but it was the Australian who bagged the first break in the next game.
That set off a run of five games in a row for the US Open champion as she began to use her power to outgun her Italian opponent. Stosur put away the set 6-1.
The momentum switched again though at the start of the deciding set as Stosur started to misfire on serve and with her groundstrokes.
Errani moved 3-0 up, only for Stosur to break back and level at 3-3.
With the wind getting up, the outcome of what had been a topsy-turvy match was on a knife-edge and, at 3-4 down, Stosur ceded serve once again, saving two break points, only to hit a forehand long and wide on a third.
Errani served out for the biggest win of her career.