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The Romanian will not play at Roland Garros this year because she is recovering from a calf injury.
The 39-year-old American went toe-to-toe with the Romanian second seed over 80 engrossing minutes on Rod Laver Arena before she was finally able to move to within two victories of a record-equalling 24th Grand Slam title.
Williams, 39, is among three Americans who have made it to the quarter-finals at Melbourne Park this year and Wednesday's clash between Jennifer Brady and Jessica Pegula has ensured that at least one from the country will be in the last four.
The world No. 2 believes the experience of playing at bio-secure tournaments last year will help her get through the 14-day quarantine.
World No. 1s Ash Barty and eight-time champion Novak Djokovic will lead the fields at the Grand Slam slated to be held from February 8-21.
Djokovic, who won the Australian Open in February for his 17th major title, is still No. 1, with Nadal at No. 2, followed by US Open champion Dominic Thiem and Federer.
Swiatek recovered from a nervy start to outclass Italian qualifier Martina Trevisan 6-3, 6-1.
Barty, 24, made the Australian Open semifinals in January but skipped the US Open and her French Open title defence citing a lack of preparation.
Swiatek crushed Halep 6-1, 6-2 to make her maiden Grand Slam quarterfinal.
Anisimova caused a huge upset last year when she comfortably beat Romanian Halep, then the defending champion, in the quarterfinals, but could not reproduce that form.