World number 58 Bertens often had the extra shot against Keys, setting up a meeting with Swiss eighth seed Timea Bacsinszky for a place in the last four.
In a match that should have started on Monday but was delayed following two days of rain at Roland Garros, Bertens broke to lead 3-2 in the opening set.
Keys broke back to force a tiebreak but she could not prevent Bertens, who won the Nuremberg claycourt tournament last month, from claiming the set after the Dutchwoman produced some stunning angled winners.
Bertens stole her opponent's serve again in the fourth game of the second set and held throughout, ending the contest on the first match point with a backhand passing shot.
Bertens said that although she felt a little sore because of the heavy balls, she adapted well to the heavy conditions on another gloomy day.
"It's really different. I think I feel it already a little bit in my shoulder. The balls are much heavier than the other days," she told reporters.
"Also my shots were not like they would have been if the sun was out but I think still I did a pretty good job today."
(Reporting by Julien Pretot,; Editing by Pritha Sarkar)
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