The Russian, seeded No.21, arrived in Paris with a lean 5-10 record in 2019, but played solidly in a 6-2, 6-3 win over Italian qualifier Jasmine Paolini.
Another victor was 22nd seed Bianca Andreescu in her first match since the Miami Open in March.
The Canadian starlet, a surprise champion at Indian Wells, suffered a shoulder injury which decimated out her clay-court season, yet returned to beat Czech Marie Bouzkova 5-7, 6-4, 6-4 in a match that was also suspended for darkness from the day before.
Kasatkina next faces Puerto Rico's Monica Puig - a straight-sets winner over Belgian Kirsten Flipkens - while Andreescu tackles Sofia Kenin for a place in the last 32.
Also advancing was Romania's Irina-Camelia Begu, who trounced China's Zhu Lin 6-1, 6-1.