With Eugenie Bouchard, Agnieszka Radwanska and Venus Williams having already been eliminated from the tournament, Wozniacki and top seed Svitolina were the remaining star names in the draw.
Svitolina withdrew with a groin injury sustained in a marathon defeat to Caroline Garcia at the China Open, while Wozniacki cited an elbow problem when she pulled out shortly before her match against Cabrera on Thursday.
"I want to apologise to everyone who came out to watch this match," defending champion Wozniacki told the crowd on court.
"I did everything I could possibly to get ready. But unfortunately I hurt my elbow during practice and there's just no way I could perform my best out here and unfortunately I have to withdraw from the tournament."
Svitolina and Wozniacki are among the eight players to have qualified for the 2017 WTA Finals, which will be held in Singapore later in the month.
Jennifer Brady set up an all-American quarter-final clash against Gibbs by upsetting Zhang Shuai in straight sets, while China's Wang Qiang will face Samantha Stosur in the last eight after triumphing 6-4 6-4 against Luksika Kumkhum of Thailand.
Cabrera will play Daria Gavrilova in Friday's quarter-finals, while sixth seed Anastasia Pavlyuchenkova faces Williams' conqueror Naomi Osaka.