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.
Barty faced a match point but the defending champion and world number one dug deep to prevail in three sets against Kristina Kucova on Thursday.
Romanian third seed Halep also needed three sets at the WTA Premier tournament in Miami.
BARTY PRODUCES COMEBACK
In her first match outside of Australia in more than 12 months, Barty survived to top qualifier Kucova in Miami.
Barty trailed 5-2 in the third set, with Kucova earning a match point at 5-3 but the Australian star saved it and won the final five games of the match to secure a spot in the round of 32.
Winner of the 2019 French Open, Barty sent down an equal career-high 15 aces in two-and-a-half hours.
"Today was really hard work, and I enjoyed every single minute of it," Barty said on court post-game. "There’s nothing like coming through a test like that, and now I get another opportunity in a couple days' time to play another tough match and test myself again."
Next up for Barty is 2017 Roland Garros champion Jelena Ostapenko, who outlasted Kirsten Flipkens 6-2 5-7 6-3.
HALEP REACHES MILESTONE
Halep brought up her 400th WTA main-draw triumph with a 3-6 6-4 6-0 victory over France's Caroline Garcia.
After dropping the opening set, Halep won 11 of the final 12 games to move through to the next round.
Halep won in one hour, 48 minutes, drastically improving on her first serve after the opening set while coming to grips with Garcia's kick serve.
"I needed time to get used to it and to get the rhythm," Halep said about Garcia's serve having improved her head-to-head record to 7-1, with Anastasija Sevastova awaiting in the last 32 after eliminating American sensation Coco Gauff.
Halep received attention on her shoulder early in the second. "I struggled with my serve," Halep said. "I struggled with my shoulder a little bit, so the attention was a little bit on the pain. [After that] I started to relax myself."
KERBER'S DOUBLE BAGEL
Three-time grand slam winner Angelique Kerber has been up and down in recently, but she flexed her muscles with a 6-0 6-0 humiliation of Renata Zarazua.
Aryna Sabalenka – the seventh seed – was another one to fend off match points before rallying 0-6 6-3 7-6 (11-9) past Bulgarian qualifier Tsvetana Pironkova in a gutsy.
Last year's French Open champion Iga Swiatek knocked off Barbora Krejcikova 6-4 6-2, while ninth seed and two-time Wimbledon champion Petra Kvitova got past Aliza Cornet 6-0 6-4.
Fifth seed Elina Svitolina survived to beat Shelby Rogers 3-6 7-5 6-3 and two-time major winner Victoria Azarenka benefited from a walkover.
