The British number one committed a flurry of errors in the opening set, in similar fashion to her opening-round match.
And she once again bounced back to win - but only after coming from three match points down in the tie-break in a marathon 97-minute second set.
She will face fourth seed Agnieszka Radwanska, who earlier beat Irina-Camelia Begu 6-3 6-3, in the last 32 on Friday.
Radwanska has already won two tournaments this year for an 11-0 record in 2013.
World number 50 Watson, 20, struggled with the extreme heat on Court 8 and did not put in her best technical showing against the 82nd-ranked Kazakh.
Three breaks of serve gave Pervak the opener and she then went 2-0 up in the second stanza - breaking to love in the first game - as Watson looked set to exit.
However her fighting spirit was to prevail: playing in her first Melbourne second round match, she began to find winners among the errors.
Such mistakes put 21-year-old Pervak 6-3 up in the breaker, only for Watson to suddenly become as solid as a rock from the baseline and save all three chances before converting her second set point.
Pervak took a long bathroom break to regroup but it did her no good as Watson moved 4-0 up in the third set within 25 minutes - despite suffering from cramp in the afternoon sun.
A love service game gave Watson a victory that seemed improbable at one stage.
"I do make it hard for myself but a win's a win, no matter how you do it," she said.
"It was a very tough one today. I wasn't sure whether I was going to get through it but I did, I kept fighting, the crowd was amazing and kept pushing me on from start to finish, and I've put myself in a position to play another match here.
"You've just got to keep fighting and fighting, because it's never over until that last point's done.
"She's a tough opponent, she makes every single ball. I didn't win a point until the eight or ninth point of the match. She wasn't going to give me anything, but you've just got to fight through and find a way."
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- Agnieszka Radwanska