Errani was once as high as No. 5 and a finalist at Roland Garros in 2012, but she has won only two Grand Slam matches since the start of 2016 and is now ranked 102nd. Bencic has been ranked No. 7 and is now 135th.
They combined for 14 service breaks in 28 games Tuesday.
Among other matches, Naomi Osaka beat Kristina Kucova 6-2, 6-3; Kirsten Flipkens edged Jennifer Brady 3-6, 6-3, 7-5; Carina Witthoeft eliminated Nicole Gibbs 6-3, 6-3; and Mandy Minella got past Kristyna Pliskova 7-6 (4), 5-7, 7-5.
Women's play began at the hard-court tournament Tuesday; the men start Wednesday.
Serena Williams, Andy Murray and Novak Djokovic all pulled out of the event with injuries.