Maria Sharapova joined Serena Williams as fourth-round casualties, leaving Azarenka a potentially easier run to the women's final. Sharapova was beaten by No. 20 Dominika Cibulkova in the match on Rod Laver Arena immediately before Azarenka defeated Sloane Stephens, the last remaining American in the singles draw.
Nadal won three tough sets against Kei Nishikori, while Murray wasted four match points in the third set — including two in the tiebreaker — before dominating the fourth set against lucky loser Stephane Robert of France.
Four-time Australian Open champion Roger Federer was never threatened in beating 2008 finalist Jo-Wilfried Tsonga 6-3, 7-5, 6-4 in a late match. He will play Murray in the quarterfinals on Wednesday.
Third-seeded Sharapova, complaining of a hip ailment, lost 3-6, 6-4, 6-1 after struggling with her serve and making 45 unforced errors. She joins Williams on the sidelines after the top-ranked American lost in three sets Sunday to Ana Ivanovic, ending 25-match winning streak.
Sharapova had spent the latter months of last year on the sidelines with a right shoulder injury, so she wasn't expecting much of herself.
"I have to look at the positives and see where I have come from in four or five months," Sharapova said. "I haven't played a lot of tennis. "
Azarenka had a 6-3, 6-2 win over No. 13-seeded Stephens, the player who beat Williams in the quarterfinals here last year and then lost to Azarenka in the semifinals under controversial circumstances. Azarenka took a medical time out when Stephens started to gain momentum in that match, then came back to advance to the final and subsequently win her second Australian Open trophy.
She wasn't getting carried away Monday with the apparent ease of the draw with Williams and Sharapova gone.
"The players who beat those players deserve all the credit because they've been better, so they are dangerous and they are competitive," Azarenka said.
Azarenka, who is on an 18-match winning streak at Melbourne Park, will next play No. 5 Agnieszka Radwanska, who beat Spain's Garbine Muguruza 6-1, 6-3 in the final fourth-round match Monday night.
Nadal beat Nishikori 7-6 (3), 7-5, 7-6 (3), though he was broken twice in the third set — getting a time violation from chair umpire Evanthia Asderaki in one of the games.
"Normally I have a good self-control on court, and it's true that the moment was a very critical one," Nadal said of the time violation.
The 13-time major winner also tripped in the first set and tumbled to the court, then had to go to the locker room to change shoes.
Nadal next plays Grigor Dimitrov, who reached the quarterfinals of a Grand Slam for the first time with a 6-3, 3-6, 6-2, 6-4 win over Roberto Bautista Agut.
Three-time Australian Open finalist Murray ended the run of Robert, wasting match points in the third set and smashing his racket after the tiebreaker before advancing 6-1, 6-2, 7-6 (6), 6-2.
"He's a fun player to watch, but not a fun player to play," Murray said of Robert's assortment of drop shots and deft groundstrokes.
The racket smashing was pure frustration.
"Losing that set was frustrating because it obviously means you're out there another 30, 40 minutes at least, when I would preferably had been in the locker room," Murray said. "My racket bit the dust. Unfortunate for it."
Murray said he's expecting a tough match against Federer, regardless of the Swiss player's recent drop in the rankings that has seen him seeded sixth here.
"I think last year you could see at periods he was struggling with his back," Murray said. "And if he's fully fit, I've said all along, he's always going to be there or thereabouts in the majors, and he'll give himself opportunities to win more because he's that good."
Federer lost to Murray in the semifinals last year in five sets after beating Tsonga in five in the quarterfinals.
"It's the same draw just one round earlier for me," Federer said. "I'm happy this time I didn't play five sets."
On Tuesday, the first of the quarterfinals begin when Ivanovic plays Eugenie Bouchard and two-time Australian Open finalist Li Na takes on Flavia Pennetta.
In men's quarterfinals, three-time defending champion Novak Djokovic plays Stanislas Wawrinka and No. 3 David Ferrer takes on No. 7 Tomas Berdych.