Williams, 36, relied on her big serve when pressed in posting a 6-2 6-3 victory under the roof at Arthur Ashe Stadium, turning away four of five break points held by the 64th-ranked Goerges.
Williams, the U.S. Open winner of 2000 and 2001, was runner-up to top seed Martina Hingis as an unseeded player in the inaugural year of Ashe Stadium and 19 years later finished the first U.S. Open day session played entirely under a roof.
"1997 was a long time ago," Williams said on court after her victory. "I'm so grateful to be here still playing, still winning matches and I can't wait for the next round."
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The elder Williams sister will next play either 26th seed Laura Siegemund of Germany or unseeded American Nicole Gibbs.
(Editing by Mark Lamport-Stokes)
