Belarusian Azarenka, who won last month's Australian Open, demolished American Christina McHale 6-0 6-0 in just over an hour.
"I felt like I came out very focused, with a good mindset and a good game plan," Azarenka said on the WTA website.
"There were a few close games, but I felt like I was really taking advantage of every opportunity I had and not letting her play her game," added the top seed who notched the sixth 6-0 6-0 win of her career.
Azarenka saved both of the break points she faced and crushed the American on serve to set up a last-eight clash with Italian sixth seed Sara Errani, who needed three sets to overcome Slovakia's Daniela Hantuchova 7-5 5-7 6-2.
The top seed is now 11-0 in 2013 having won the Australian Open. She will retain the number one spot in the WTA rankings if she reaches the final and Williams fails to reach the semis.
Fifteen-times Grand Slam champion Williams crushed Pole Urszula Radwanska 6-0 6-3 while Russian third seed Maria Sharapova overpowered Czech Klara Zakopalova 6-3 6-3.
Fourth seed Agnieszka Radwanska beat former world number one Ana Ivanovic 6-1 7-6 and will next face Caroline Wozniacki, another former world number one, who defeated German Mona Barthel 7-6 6-3.
"It wasn't great tennis out there today - the wind was a huge factor," Wozniacki said. "To be honest it was just about getting the ball in play today.
"I was just trying to get it in and hoping for the best. It was a struggle, but I got through it," added the Dane.
Australian Sam Stosur reached the last eight with a 6-4 7-5 win over Russian Svetlana Kuznetsova.
Czech Petra Kvitova completed the quarter-final line-up after recovering from a slow start to overcome Russian Nadia Petrova 4-6 6-4 6-3.
10-Caroline Wozniacki (Denmark) beat Mona Barthel (Germany) 7-6(6) 6-3
4-Agnieszka Radwanska (Poland) beat 13-Ana Ivanovic (Serbia) 6-1 7-6(6)
1- Victoria Azarenka (Belarus) beat Christina McHale (U.S.) 6-0 6-0
8-Samantha Stosur (Australia) beat Svetlana Kuznetsova (Russia) 6-4 7-5
6-Sara Errani (Italy) beat Daniela Hantuchova (Slovakia) 7-5 5-7 6-2
3-Maria Sharapova (Russia) beat Klara Zakopalova (Czech Republic) 6-3 6-3
2-Serena Williams (U.S.) beat Urszula Radwanska (Poland) 6-0 6-3
7-Petra Kvitova (Czech Republic) beat 11-Nadia Petrova (Russia) 4-6 6-4 6-3