Venus Williams, 35, produced a vintage power-game performance to beat up-and-coming 18-year-old Bencic 6-3 6-4 in a third-round match-up between the oldest and youngest players left in the women's draw.
Williams, who played her first U.S. Open in 1997, the year 12th-seeded Swiss was born, won 83 percent of her first serves, and blasted 31 winners, including 15 from the forehand side.
Bencic, who beat 2015 Australian, French and Wimbledon champion Serena Williams on her way to winning in Toronto last month, had been considered a Flushing Meadows contender since rising from 32nd to 12th in the world this year.
Venus Williams said world number one Serena offered some advice on playing Bencic, but added that Williams family revenge was not on her mind.
"Whoever is across the net, that's who I want to beat," she said.
Venus, who won the second of her two U.S. Open titles 14 years ago, had battled to a pair of three-set wins and spent nearly five hours on court to reach the third round but made quick work of Bencic.
"She was just too good I think today," said the Swiss.
The 23rd-seeded Williams allowed just one break point in the 78-minute match.
Bencic converted that one chance in the fourth game of the second set on her way to a 4-1 lead before Venus ran off five games in a row to set up a Round of 16 meeting with Estonia's Anett Kontaveit, who beat American Madison Brengle 6-2 3-6 6-0.
"I just wanted to stay aggressive and keep my errors down," said Venus, who had also beaten Bencic in straight sets in their three previous encounters.
Joining the seven-times grand slam singles winner in the fourth round was Russian 13th seed Ekaterina Makarova, a semi-finalst at Flushing Meadows last year.
Left-hander Makarova, despite cramps that hit her during the last game of her match, advanced with 6-3 7-5 victory over 17th seed Elina Svitolina of Ukraine.
Makarova next meets France's Kristina Mladenovic, a 6-2 6-3 winner against Russian Daria Kasatkina, who had replaced third seed Maria Sharapova in the draw after she withdrew with a leg injury.
(Editing by Mark Lamport-Stokes)
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