The 31-year-old 13th seed battled back from 2-5 down in the final set to defeat US 18th seed Sloane Stephens 6-7 (1/7), 6-2, 8-6 and set up a last-16 clash with defending champion Serena Williams.
But the Russian was hit with a code violation for coaching early in the final set which prompted a bitter exchange with umpire Marijana Veljovic.
"I'm just doing my job," said the official.
"Well, you're not doing it very well," responded Kuznetsova, the 2004 US Open champion and 2009 French Open winner.
"It's really frustrating," said the three-time quarter-finalist.
"She warned me but I said, 'did you hear what he said?' I told her he talks a lot.
"He just said, 'great job, keep going'. It's very frustrating. I lost a little momentum after that but I managed to fight back.
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