It was Stosur's first appearance in the competition after she replaced defending champion Petra Kvitova, who was forced to withdraw due to a viral illness.
There were signs of rustiness from the Australian as she dropped the first game on serve, but she found an immediate response by breaking Errani in the next.
However, a pair of unforced errors in the seventh game gifted Errani a second break, and the hard-working Italian went on to win the opener.
Stosur played some of her best tennis to level the scores by claiming the second set, breaking three times and hitting a succession of trademark forehands to take her winners count to 32 compared to Errani's 10.
However, the former US Open champion could not build on this momentum as Errani showed displayed great determination to win the third set without dropping a game.
The seventh seed faces Agnieszka Radwanska in a winner-takes-all match on Friday, with the victor progressing to the last four.