Williams - stuck on 23 majors since winning the 2017 Australian Open in pursuit of Margaret Court's all-time record - was pushed to the limit by Russian teenager Potapova but prevailed 7-6 (7-5) 6-2 in Melbourne on Friday.
Next up for seven-time Australian Open winner Williams is seventh seed Aryna Sabalenka.
Williams - who has played and won more matches than any other woman in the tournament's history - had only lost four times at a slam to a player ranked outside the top 50 and twice to players outside the top 100.
But she was challenged by Potapova in an error-riddled and difficult opening set on Rod Laver Arena, where the unheralded Russian took the match to her superstar opponent.
Williams hit 22 unforced errors to just 12 winners, faced 11 break points as she was broken twice, and saved a pair of set points en route to a tense tie-break.
Maintaining her challenge, Potapova led 4-2 and 5-3 in the tie-break, but Williams was up to the task - closing out the set in 61 minutes.
Potapova, who only broke the American's serve on one occasion in last year's opening-round loss at Melbourne Park, was not deflated after the tie-break and broke Williams in the first game of the second set.
But like the first set, Potapova was unable to hold serve and Williams took full advantage - breaking twice to secure a 16th fourth-round berth in Melbourne.