Stosur, the former world number four, was the first of the pair to advance to the semi-finals after beating Danka Kovinic 6-3 6-2 in 66 minutes.
“I feel like today was definitely the best I’ve hit the ball and played all week,” Stosur told the official WTA website.
“To go through a match not facing a break point, it definitely shows I’m serving well.”
Stosur, the second seed, now faces Anna Karolina Schmiedlova, who won 15 points in a row at one point to beat Annika Beck 6-2 6-3.
The seventh seed from Slovakia has now won eight successive matches – five to win in Bucharest last week and three in Bad Gastein – and said: “It’s been a good few weeks for me and I was really happy with the way I played today as well.”
Top seed Errani followed Stosur into the semi-finals with a hard-fought 6-3 3-6 6-2 win over Daria Kasatkina as the Russian qualifier fought back from losing the first set and being a break down in the second.
Errani’s semi-final opponent will be third seed Karin Knapp, who had a busy day by first beating Johanna Larsson 6-3 6-3 in the second round before dispatching Polona Hercog 7-6 (7/5) 7-5 in the last eight.