Svitolina outlasted Daria Gavrilova of Australia 4-6 6-0 6-1 and the sixth seeded Ostapenko turned back Brazil's Beatriz Haddad Maia 6-2 7-6(2).
Konta, the 11th seed, crushed Elise Mertens of Belgium 6-2 6-1 to set up a fourth round meeting with the older Williams sister.
The tournament looked like losing yet another big name when Venus fell behind 4-1 in the third set on centre court at Crandon Park in south Florida.
However, she remained more poised on the big points and eventually won on serve when Bertens could only tap the ball back across the net, allowing the 37-year-old American to put away the winner.
"I'm not sure," the eighth seed said on court when asked how she had pulled off the victory in a match that had 15 breaks of serve.
"I had a lot of chances, she had a lot of chances. It didn't look good sometimes but that's why you keep playing to the last point, and why I love this game."
Venus's survival to fight another day came as the women's draw remained wide open after the earlier ousting of her sister Serena along with top seed Simona Halep and last week's Indian Wells winner Naomi Osaka.
Konta mixed 23 winners and 17 unforced errors against Mertens while the Belgian, a 2018 Australian Open semi-finalist, was slowed by 26 unforced errors and had only 11 winners.
"I have really great memories from last year," said Konta in an on-court interview. "I want to thank everyone for coming out and making my return to the center court a special one."
Also winning were Australian Ashleigh Barty, the 21st seed, who took down Croatia's Petra Martic 6-4 6-3, and Puerto Rico's Monica Puig who beat Maria Sakkari of Greece 6-3 7-5.
(Reporting by Andrew Both in Cary, North Carolina; Additional reporting by Gene Cherry; Editing by Clare Fallon/Nick Mulvenney)
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