The seven-time Grand Slam champion is the oldest woman in the draw at 39. She has made her deepest Cincinnati run since reaching the 2012 semifinals.
Bertens, seeded fifth, rallied from deficits of 3-0 and 5-2 in the third set to force a tiebreaker. The 2-hour, 17-minute match ended when she sailed a forehand wide.
Two-time Australian Open champion Victoria Azarenka, who won Cincinnati in 2013, reached the second round after 12th-seeded Belinda Bencic retired because of an injured left foot while trailing 6-4, 1-0.
The Associated Press contributed to this report.