Doi, who advanced to the fourth round of Wimbledon earlier this month, cruised past Xinyun Han 6-0 in a 24-minute first set but fell 2-6 in the second. Doi started the third set by winning three straight games en route to 6-2.
Cornet, the world's No. 51st-ranked player, beat Jennifer Brady 3-6, 6-4, 6-4 in a match lasting 2 hours, 43 minutes and will face Saisai Zheng.
Nicole Gibbs topped fellow Stanford standout Carol Zhao 6-2, 4-6, 6-3 in the first evening match. Gibbs, who won NCAA singles titles in 2012 and 2013, advances to play No. 4 seed CoCo Vandeweghe on Thursday. Gibbs is partnered with Zhao in the 16-team doubles field.
Top-seeded Venus Williams, a recent Wimbledon semifinalist and a four-time Olympic gold medalist, plays her first match of the women-only tournament on Wednesday night.