Roared on by the Arthur Ashe Stadium crowd, Williams, who won two titles at Flushing Meadows in 2000 and 2001, overcame serve problems to set up a meeting with gritty Spaniard Carla Suarez Navarro.
She has now reached the last 16 in every grand slam this year, a feat she last achieved in 2010.
The 37-year-old managed the only break in the opening set to take control of the match.
After a couple of exchanges of breaks in the second, Williams stole her opponent's serve again to close it out on the first match point when Sakkari's forehand sailed long.
"I'm playing at home, I've got a lot to accomplish here, and I'm happy to get the win," she said, before adding she was expecting a similar game style from Suarez Navarro as Sakkari.
"She's feisty. I know I have to come out, play well, have you all (the crowd) behind me and get a win."
The question on everyone lips, however, was whether her younger sister Serena, the 23-times grand slam champion, had given birth to her first child.
Venus refused to answer the question and a reporter was cut off by a moderator after she asked several questions related to the expected birth.
(Reporting by Julien Pretot; Editing by Ed Osmond/Greg Stutchbury)
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