Ranked 272nd and playing as a qualifier, Bouchard had three match points at 5-4 in the second set before Kuznetsova forced a deciding set. The Russian was one of four seeded players knocked out in Thursday's second-round matches.
Bouchard, who reached the Wimbledon final in 2014 but lost to Petra Kvitova, will next face Danka Kovinic in the quarterfinals. Kovinic eliminated sixth-seeded Alison van Uytvanck 6-3, 6-4. Bouchard last went on such a run in 2018 when she won six straight from qualifying to the semifinals in Luxembourg.
Also, second-seeded Rebecca Peterson swept into the quarterfinals when her second-round opponent Margarita Gasparyan retired with a knee injury at 3-2 down in the first set. Third-seeded Polona Hercog reached the quarterfinals by beating Jasmine Paolini 7-5, 6-4. Tereza Martincova produced another surprise result with her defeat of No. 4 Caroline Garcia 6-1, 6-4 in 1 hour, 9 minutes. Patricia Maria Tig knocked out eighth-seeded Misaki Doi 6-2, 6-0.