The world number one was pushed to the wall by the 27-year-old Russian, who played more like a two-time Grand Slam winner than her current world number 75 ranking.
Azarenka established her dominance in the second set, however, to set up a semi-final against American teenager Sloane Stephens, who sent shockwaves through the tournament by upsetting Serena Williams 3-6 7-5 6-4.
"It would be silly to say I'm not excited, I'm not nervous," said Azarenka. "I think the person who is never nervous is a really scary person. I mean, is a robot really.
"Everybody has emotions. It's just a matter how well you can ... control them for you to feel good before you go on court or when you're on the court."
Kuznetsova and Francesca Schiavone played in the longest women's grand slam singles match in 2011 at Melbourne Park, lasting four hours and 44 minutes, and the Russian appeared determined to drag Azarenka into another draining epic.
Former US and French Open champion Kuznetsova raced to a 4-1 lead, with her sliced serve to her opponent's forehand proving a handful for the Belarusian.
But Azarenka moved up a gear to level at 4-4, and captured the decisive break at 5-5 before sealing the first set when Kuznetsova missed a backhand.
Kuznetsova spent half of 2012 recovering from a knee injury and suffered a fright when the same knee jolted during a serve in the first game of the second set.
"I was really scared because I almost broke it again the same way I did it (last year)," she said. "My knee went backward, the same way, exactly ... I was a little bit freaking out because I didn't want to have another time off."
The 77-minute first set lasted 20 minutes longer than Azarenka's entire match against Russian Elena Vesnina in the previous round, but she roared through the second in half an hour.
"I think it was a lot of pressure there, a lot of, as I said, back and forth," Azarenka said. "It was important to take the opportunities to kind of make a big statement.
"I think I did that and I turned things around into my own way," she said.
Azarenka has now qualified for her fourth semi-finals appearance in the last five Grand Slams, but despite the absence of Williams, may still need to battle to keep her world number one ranking from Maria Sharapova at Melbourne Park.
Sharapova plays her semi-final against China's Li Na on Thursday.
"I think Sloane is a very, very talented girl," Azarenka told reporters before the all-American quarter-final.
"I think over the last couple years you see her development, you know, as a tennis player. Her game has come together. She's all-around player, very competitive.
"I think she has a good potential. It's just a matter of her, you know, getting all those details in place and really wanting to achieve great things."