The Czech will next face either Croatian 31st seed Petra Martic or home hope Kristina Mladenovic for a spot in the second week.
Pliskova, a former Roland Garros semi-finalist, arrived in Paris full of confidence after lifting the Italian Open title less than two weeks ago and she wasted no time in swatting aside Kucova in under an hour.
Svitolina to face Muguruza
Ukrainian Elina Svitolina will face 2016 champion Garbine Muguruza in the women's third round after the ninth seed's compatriot Kateryna Kozlova pulled out of the tournament before their match with a viral infection.
Spain's Muguruza, seeded 19th this year, eased to a 6-4, 6-1 victory over Swede Johanna Larsson to advance.
Svitolina, a two-time French Open quarter-finalist, had beaten seven-time Grand Slam champion Venus Williams in her opening tie.
Last year's runner-up Sloane Stephens also eased through with a 6-1, 7-6 (7/3) win over Sara Sorribes Tormo on Court Philippe Chatrier.
The former US Open champion, who lost to Simona Halep in the 2018 French Open final, will play Slovenia's Polona Hercog next after she edged out American Jennifer Brady 6-3, 6-7 (8/10), 6-4.
Seventh seed Stephens has now reached at least the third round at Roland Garros on each of her last seven visits to the French capital.