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.
In a match that lasted 2 hours, 34 minutes, Dimitrov fought off two match points in the eighth game of the third set and forced a tiebreaker. Wawrinka won the match with an ace.
Wawrinka next faces qualifier Andrey Rublev, who advanced with a 6-7 (4), 6-4, 6-2 victory over 15th-seeded Nikoloz Basilashvili.