Carre?o needed three hours and one minute to prevail over the 147th ranked Slovakian in their first career encounter, reports Efe.
The Spaniard struggled on serve during the first set, as he conceded his service game twice, to fall a set behind.
Without conceding his service game once, Carre?o made the most of two out of five break points he created in the second set to draw level.
The third set saw a total of five service breaks, with Carre?o grabbing the decisive break to move a set away from advancing to the second round.
Kovalik did not face a single break point in the fourth set, but he could not seize on the lone opportunity he created as the set went to a tie break that Carre?o sealed.
A 2017 French Open quarterfinalist, Carre?o is set to play either Argentina's Federico Delbonis or Brazil's Thomaz Bellucci.
In other French Open action on Sunday, Grigor Dimitrov of Bulgaria, world No. 5, defeated Mohamed Safwat of Egypt 6-1, 6-4, 7-6 (7-1).
Despite his loss, Safwat had the good fortune to replace injured Viktor Troicki of Serbia, becoming the first Egyptian player to make it to the main draw of any Grand Slam championship since 1996.
Next up for Dimitrov, fourth seeded, is the United States' Jared Donaldson, who defeated Nicolas Jarry of Chile 6-3, 4-6, 3-6, 6-3, 6-4 in two hours and 50 minutes.
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