Pegula, who beat the fourth-seeded Karolína Pliskova for the fourth time this year on Thursday, was unable to follow up against the seventh-seeded Azarenka, losing 6-2, 5-7, 6-4. The Belarusian was playing her first grass-court quarterfinal since Wimbledon in 2015.
The 106th-ranked Samsonova needed almost 2½ hours to shock Keys 7-6 (4), 2-6, 7-6 (0).
Samsonova plays Azarenka in her second career semifinal on Saturday.
It will be the 31-year-old Cornet's first semifinal on grass.
Bencic is through to her fifth on grass. The Swiss hit nine aces against Alexandrova, who replied with 15 of her own. Bencic won the first two points in the tiebreak and held on to her lead.