
Russian teenager Andreeva, who received a first-round bye, will face either German Ella Seidel or Romanian Jaqueline Cristian for a quarter-final place after Daria Kasatkina pulled out with a hip injury.
In-form Czech youngster Sara Beljek, who won a title in Abu Dhabi earlier this month, also withdrew with an abdominal injury, handing Bencic a spot in round three.
American fourth seed Pegula cruised through with a 6-4, 6-0 thrashing of Varvara Grachev and will next play compatriot Iva Jovic.
The tournament in Dubai, one of the 10 WTA 1000 competitions, has been severely hit by the withdrawals of world number one Aryna Sabalenka and six-time Grand Slam champion Iga Swiatek.
Second seed Amanda Anisimova had also been given a walkover on Monday after her second-round opponent Barbora Krejcikova pulled out.
Later Tuesday, American Coco Gauff starts her campaign against Russian Anna Kalinskaya.