Wozniacki is bidding for a fourth title this year and will play either former No. 1 Jelena Jankovic or Andrea Petkovic of Germany.
Kucova, plucked from qualifying after Yanina Wickmayer withdrew with an injured knee, broke Wozniacki once but struggled to hold her own.
Vera Zvonareva also advanced, rallying to defeat Anastasia Pavlyuchenkova 3-6, 6-4, 6-2 in an all-Russian match.
The second-seeded Zvonareva remains on course to add to her Doha title this year after struggling past Pavlyuchenkova on the indoor clay court at Porsche Arena.
Also in the second round, Agnieszka Radwanska of Poland upset French Open champion Francesca Schiavone 6-1, 6-3, Sabine Lisicki downed Australian Open runner-up Li Na 6-4, 7-5, and Kristina Barrois defeated eighth-seeded Marion Bartoli 6-4, 6-2.
Barrois and Lisicki are the first Germans to reach the last eight in Stuttgart since Anke Huber 10 years ago.
The third-seeded Schiavone won her first four matches against Radwanska, but has lost to the Pole both times this year.
Barrois landed 81 percent of first serves against the 12th-ranked Bartoli, the highest-ranked player she has beaten.
“I was a total outsider,” the 79th-ranked Barrois said. “This is a great success for me.”
Lisicki broke to lead 6-5 in the second set and served out the match against the sixth-seeded Chinese.
The German was once ranked as high as No. 22, but was out for nearly half a year with an injured left ankle and has dropped to No. 155.
Like Barrois, Lisicki needed a wild-card entry for the tournament.
In Fez, Morocco, Moroccan wild card Nadia Lalami upset top-seeded Aravane Rezai of France 2-6, 7-6 (5), 6-4 to advance to the quarterfinals of the Grand Prix SAR on Wednesday.
The 497th-ranked Lalami broke twice in the final set and served out the match on her first opportunity to ensure Rezai still hasn’t won consecutive matches since July.
American Melanie Oudin upset third-seeded Lourdes Dominguez Lino of Spain 4-6, 6-1, 6-2.
Rezai, who had 11 double faults, came back from 5-1 down in the second set only to lose the tiebreaker at the clay-court tournament.
The Frenchwoman has won four WTA titles but has been hampered by personal problems recently. She hired coach Alexia Dechaume this week in a bid to revive her career and get ready for the French Open next month.
Lalami will face either Eleni Daniilidou of Greece or Kirsten Flipkens of Belgium.
Greta Arn of Hungary overcame problems with her serve to beat Kristyna Pliskova of the Czech Republic 6-4, 6-4.
The fourth-seeded Arn, who won her second career title in Auckland in January, had eight double faults but managed to save all four break points she faced.
Arn will face seventh-seeded Simona Halep of Romania, who beat Alla Kudryavtseva of Russia 6-2, 6-4. Halep went from qualifying all the way to the final last year, still the best result of her career.
Slovenia’s Polona Hercog needed almost three hours to beat South Africa’s Chanelle Scheepers 6-7 (6), 7-6 (1), 6-2.
