Kenin is the top seed in France this week and belatedly secured her first WTA Tour victory since a major breakthrough in Melbourne.
The American had fallen at the first hurdle in both Qatar and Dubai in recent weeks, but she had no such troubles against Vitalia Diatchenko, advancing 6-4 6-3.
A pair of early breaks in the opener saw Kenin into a 4-1 lead, although opportunities for both players punctuated the contest.
The second set began with three straight games going against the serve, before Kenin settled the match at the second attempt with her sixth successful break point from 18 attempts.
Back on the winning trail, Kenin levelled her career record against Diatchenko at 1-1 after a second-round defeat at Wimbledon in 2018.
The only other round-of-32 clash saw France's Oceane Dodin overcome Mandy Minella in straight sets.
Meanwhile, Kenin's fellow seeds Caroline Garcia - a 7-5 6-2 winner against Ysaline Bonaventure - and Alison Van Uytvanck - triumphing over Viktoriya Tomova - enjoyed victories as they played last-16 matches.
But home hopeful Cornet fell short of the quarter-finals in a 4-6 7-6 (7-5) 6-2 reverse at the hands of Camila Giorgi.
Cornet led by a mini-break three times in the second-set tie-break, before Giorgi edged her opponent out and then dominated the decider.