The tournament is the first in five months since the Tour was suspended due to the COVID-19 pandemic and Martic said she felt the effects of not playing competitive tennis for so long.
Top seed Petra Martic held her nerve in two tiebreaks to beat Belarusian qualifier Aliaksandra Sasnovich 7-6(5), 7-6(3) to seal a place in the Palermo Ladies Open semi-finals on Friday.
She next faces 24-year-old Estonian fourth seed Anett Kontaveit who hit 57 winners to end Italian teenager Elisabetta Cocciaretto's dream run with a 6-1, 4-6, 6-1 win.
Martic, having come through a gruelling three-setter with Liudmila Samsonova on Thursday, was forced to concede a break in the opening set against Sasnovich before fighting back to take it to a tiebreak in which she prevailed.
The Croatian then established a 4-1 lead in the second set and looked to be headed for a straightforward victory before Sasnovich clawed her way back, saving three match points until Martic converted the fourth in the tiebreak.
"I had to adjust from my match yesterday. The points were longer and it was more physical," Martic told reporters in a virtual news conference.
"It was a tough two sets, but I'm really happy that, even though things weren't going my way at the end of the second set, I managed to pull through.
The tournament is the first in five months since the Tour was suspended due to the COVID-19 pandemic and Martic said she felt the effects of not playing competitive tennis for so long.
"I had two tough matches and my body's holding up well. I have no major issues," Martic added. "Just maybe soreness that I'm sure we all have in the first tournament back. It feels nice to be reminded of how it feels to be that kind of tired."
Wildcard Cocciaretto had upset sixth seed Donna Vekic in the second round but struggled against Kontaveit as she lost the first set. The Italian recovered to win the second but Kontaveit raised her level to claim the final four games in the third.
The other semi-finalists are Fiona Ferro who beat Sara Errani 6-4, 6-1 and Camila Giorgi who defeated Yastremska 4-6, 7-6(5), 6-3 in the longest match of the day. She won in 2 hours and 52 minutes to enter her 2nd WTA claycourt semi-final.
