Stosur has the ability to frustrate and thrill in equal measure, but the Australian was at her best in a quick-fire 6-1 6-0 victory over Koukalova in their second-round contest.
The second seed needed just 50 minutes to get the job done. Koukalova won her first service game to make it 1-1, but from there lost 11 straight games as Stosur ran riot.
Top seed Errani was a 6-4 6-4 victor over Sasnovich in round one at the clay-court event in Austria.
Errani broke in the first game of the match and, although the Italian surrendered that advantage in game six, she earned another break for a 5-4 lead and served out the opener to love.
There were five breaks of serve in a back-and-forth second set, but again Errani made the crucial break in game nine and took the victory on her second match point.
Also in round one, seventh seed Anna Karolina Schmiedlova defeated 2011 runner-up Patricia Mayr-Achleitner 3-6 6-2 6-0, while Johanna Larsson - winner of the Swedish Open last week - defeated Stefanie Vogele 6-2 6-2.
Fifth seed Lucie Hradecka did not fare as well, though, as she exited 6-3 6-2 to Polona Hercog.
In other round-two matches, Danka Kovinic defeated lucky loser Richel Hogenkamp 6-3 6-1, and Annika Beck overcame Denisa Allertova 6-2 6-3.