Zanevska twice found herself trailing 3-0 in the battle between two first-time WTA singles finalists, but rallied to win 6-4 7-6 (7-4) on the clay in Gydnia on Sunday.
World number 165 Zanevska had never played in a quarter-final on the main tour before making the last four in Lausanne last week and the in-form Belgian now has a first title to her name.
The 27-year-old was granted a special-exempt entry into the Gdynia main draw due to her run in Switzerland and is now projected to re-enter the top 120 in the rankings.
Kucova was left to reflect on what might have been, having started both sets so strongly before failing to press home her advantage in a topsy-turvy championship match.
The Czech, ranked 150th, saved four match points before forcing a tie-break but was beaten in an hour and 51 minutes.
World number two Kontaveit recorded a 6-3 6-2 victory over Peterson on the clay in Germany, setting up a last-eight meeting with eighth seed Andrea Petkovic.
The Estonian's win was her third over Peterson and maintained her 100 per cent career record against Swedish opponents, taking her to seven such successes.
Kontaveit was joined in the quarter-finals by fellow seeds Aliaksandra Sasnovich, who beat Aleksandra Krunic 6-3 6-2, and Maryna Zanevska, who downed Alexandra Cadantu-Ignatik 6-4 6-1.
They will face off for a semi-final spot on Thursday, while Anastasia Potapova will meet 2021 French Open champion Barbora Krejcikova after beating Maria Carle in straight sets.
At the Palermo Ladies Open, meanwhile, third seed Zhang withdrew from her last-16 clash with home favourite Jasmine Paolini before taking to the court, citing an unspecified injury.
Fellow seeds Sara Sorribes Tormo and Anna Bondar both reached the quarter-finals with straight-sets wins, however, with the pair set to face off for a semi-final berth on Friday.
The American beat Maryna Zanevska 6-3 7-6 (7-4) on Saturday, hitting 35 winners and 13 aces on the way to victory.
Parks has risen through the ranks in the last year, and explained how a laid back attitude has been the secret to her success in France.
"I think the key was just staying focused, and taking my time out there," Parks said after her win. "Every time I got ahead of myself, I would kind of lose points, so I would tell myself to slow down.
"This whole week, I'm kind of relaxed this week and just kept playing my game and doing me."
The 22-year-old will face Garcia on Sunday after she eased to a 6-2 6-2 win against Camila Osorio in just an hour and 12 minutes.
At the Thailand Open, Lesia Tsurenko will take on China's Zhu Lin in the final after the Ukrainian came past top seed Bianca Andreescu.
The Canadian retired hurt in the second set, though Tsurenko already led 7-5 4-0.
"The way [Andreescu was] playing every point, like she has an idea in every point, so of course she's an amazing player," Tsurenko said. "It was a big challenge for me to handle this, and I was just saying to myself that I have to keep fighting, and this is probably the main key that worked today for me."
Zhu defeated compatriot Xinyu Wang 6-2 6-4 to advance to Sunday's final.
The French top seed looked to be in trouble as Belgian Van Uytvanck eased to the first set, before Garcia found her feet to come back and overwhelm her opponent 2-6 6-0 6-1.
She will face Jasmine Paolini in the last eight after the Italian took just over an hour to dispose of Erika Andreeva 6-2 6-3.
Second seed Zhang Shuai is out though after falling to another Belgian in Maryna Zanevska 6-2 6-0.
Fifth seed Anastasia Potapova will face Zanevska next after her 6-4 4-6 6-3 win over Clara Burel, while seventh seed Danka Kovinic is also through after beating Anna Bondar 4-6 6-2 6-3.
At the Thailand Open, top seed Bianca Andreescu had no problems against Anastasia Zakharova, easing through to the quarter-finals 6-2 7-5.
The Canadian will face Marta Kostyuk next after the Ukrainian fifth seed defeated Nao Hibino 6-3 7-6 (9-7).
Sixth seed Tatjana Maria also progressed 6-0 6-4 against Dayana Yastremska, as did Heather Watson as she fought hard to see off Han Na-lae 6-3 5-7 6-4.
