Azarenka needed less than an hour to see off unseeded Russian Elena Vesnina 6-1 6-1 to book a last-eight clash with Kuznetsova.
Kuznetsova showed she was free of the leg injury which ruined her 2012 season when she bounced into the quarter-finals of the Australian Open.
The former French and US Open champion upset 10th seeded-Dane Caroline Wozniacki 6-2 2-6 7-5 on Monday in a gruelling encounter lasting two hours and 28 minutes.
Kuznetsova plummeted outside the top 50 last year for the first time in 11 years and came into the Open unseeded.
She had lost to Wozniacki four times in succession before her return from injury this year, but did beat the former world No.1 for the first time since 2009 in the recent Sydney International.
Kuznetsova started strongly against Wozniacki, sealing an early service break in the first set and taking it 6-2 in 43 minutes.
Wozniacki was quick to respond, levelling the scores before the pair fought their way to 5-5 in the decider.
The Russian dominated the 12th game and wasted no time capitalising after running up two match points against Wozniacki, whose returns had become more erratic.
"I'm very happy with the result but disappointed with the second set because I got stuck in there a little bit," said the Russian.
Kuznetsova said she was now refreshed and fit after becoming jaded last year.
"I broke a little bone in my knee and was on crutches for about two months," she said.
"I had been a little bit tired travelling over 10 years but I feel a lot fresher now."