After dropping serve in a long fourth game which went to deuce 10 times, Azarenka recovered to dominate the rest of the match against Kuznetsova, a two-time major winner who was floating dangerously in the draw with a No. 75 ranking as she recovers from a knee injury.
Azarenka will meet the winner of Wednesday's later quarterfinal between Serena Williams and 19-year-old American Sloane Stephens. Williams has lost only one match since a first-round exit at the French Open in a streak that included back-to-back major titles at Wimbledon and the U.S. Open as well as the Olympic gold medal and season-sending WTA Championship.
She beat Azarenka all five times they played last year, and is 11-1 in career head-to-heads against the 23-year-old Belarusian.
Azarenka's American rapper friend, Redfoo, returned from a concert in Malaysia to attend Wednesday's quarterfinal match.
Wearing a red sleeveless T-shirt that read "Keep Calm and Bring Out the Bottles," the name of his next single, Redfoo stood, clapped and yelled "Come on, Vika!" during the tight first set.
Asked if it helped to have her No. 1 fan wearing a keep calm logo, Azareka said "I was looking more at the part that says 'Bring out the bottles.'"
Of her game, she added, "I'm just glad I could produce my good tennis when it was needed."
In the men's quarterfinals, 17-time major winner Roger Federer was playing No. 7 Jo-Wilfried Tsonga in a night match and U.S. Open champion Andy Murray was to meet Jeremy Chardy of France.
Novak Djokovic is already through the semifinals after his 6-1, 4-6, 6-1, 6-4 win over fifth-seeded Tomas Berdych on Tuesday night.
The 2-hour, 31-minute victory took exactly half the time of his five-set, fourth-round win two nights previously against Stanislas Wawrinka.
"It was a great performance. I was hoping to have a shorter match ... just not to go over 5 hours," Djokovic said, in a comparatively subdued mood Tuesday after a considerably more routine victory. "It's always going to be tough against Tomas; he's an established player. He has a big game, big serve. He can compete against anyone on any surface."
In the semifinals, Djokovic will meet No. 4-seeded David Ferrer.
Ferrer survived a quarterfinal battle with fellow Spaniard Nicolas Almagro. Almagro had three chances to serve for the match, but Ferrer broke back each time and went on to win 4-6, 4-6, 7-5, 7-6 (4), 6-2.
Maria Sharapova had a 6-2, 6-2 quarterfinal win over Ekatrina Makarova earlier Tuesday, and has conceded only nine games in five matches — a record in Australia.
"To be honest, those are not the stats you want to be known for," Sharapova said.
She'll play Li Na, who reached the semifinals for the third time in four years at Melbourne Park after beating Agnieszka Radwanska 7-5, 6-3.