Williams, the third seed at the WTA Premier tournament, progressed after close to three hours on court in New York on Tuesday (AEST).

Williams was taken to the brink before prevailing 7-6 (8-6) 3-6 7-6 (7-0) against Arantxa Rus after two hours, 49 minutes.

A 23-time grand slam champion, Williams served 14 aces – including four in the final set – but was pushed by Rus.

Williams served for the match at 5-3 before losing three straight games, Rus also unable to serve it out before the American advanced.

Awaiting Williams, a two-time winner of the event, in the last 16 is Maria Sakkari after the Greek 13th seed beat Yulia Putintseva 6-4 7-6 (11-9).

Osaka was also pushed before progressing to the last 16 at the event, which is usually held in Cincinnati.

After levelling the match, Osaka raced into a 4-0 lead in the deciding set on her way to victory.

Osaka will next face 16th seed Dayana Yastremska, who recorded a 7-6 (7-5) 4-6 6-3 win over Bernarda Pera.

Aryna Sabalenka (five), Johanna Konta (eight), Anett Kontaveit (12) and Elise Mertens (14) were the other seeds to advance.

However, Keys and Kvitova were among the departures.