Top seed Williams and men's champion Wawrinka were both upset in early rounds at last year's French Open, but ensured there were no such repeats on Melbourne's blue hardcourts against the same Spanish opponents on Monday.
Battling a cough, Williams overcame the hard-hitting Garbine Muguruza, who beat her in the second round at Roland Garros, 2-6 6-3 6-2, while Wawrinka was also forced to fight to beat Guillermo Garcia-Lopez in four sets.
Garcia-Lopez dumped Wawrinka out of the French Open last year, ensuring the Swiss became the first Australian Open champion to lose in the first round at Roland Garros since Petr Korda in 1998.
Williams has struggled in her last three Australian Opens, failing to advance past the quarter-finals as she self-destructed against lower-seeded opponents.
After they traded the first two sets, Williams somehow managed to hold off six break points in her first service game of the decider, which ended the 21-year-old's resistance and she romped to a victory.
"I had to play the best match of the tournament or else I was going to be out," Williams told reporters. "When I have to go up a level, I have to. I can't afford to stay at the same level or I will be where I was at the French Open."
Williams will now meet last year's beaten finalist Dominika Cibulkova who outslugged twice champion Azarenka in the first match under the closed roof on Rod Laver Arena due to light rain that greeted fans on Australia Day.
Cibulkova proved that last year's run to the final was not a fluke in her 6-2 3-6 6-3 victory over the 2012 and 2013 champion with her all energy style, constantly buzzing around court, hitting big forehands and putting Azarenka on the back foot.
"I don't have a big serve obviously ... so I just put a lot of energy in my tennis (and) I go for my winners," the 11th seed said. "Today was extremely, extremely good match from my side."
It was the Belarusian's earliest exit at Melbourne Park since she lost at the same stage in 2011 to China's Li Na, though after an injury-plagued 2014 the unseeded 25-year-old was pleased with her progress in her comeback.
"I think there are a lot of the positive things to take from here. It's a good start," she said. "But I need to be realistic a little bit and keep working hard and try to sharpen my game."
The early morning rain also momentarily affected Wawrinka's match on the newly refurbished Margaret Court Arena with water dripping onto the court in the first set, just after the 24-times grand slam winner Court had officially opened it.
Wawrinka was pushed by Garcia-Lopez before he eventually moved into the final eight after he overcame a 5-0 deficit and fended off five set points in the fourth set tiebreak.
"It was a really really tough battle," Wawrinka said.
"We played a big match at the French Open and it is always tough to play him ... and I'm just glad to get through."
American teenager Madison Keys joined Williams in the quarter-finals after she hammered namesake Madison Brengle 6-2 6-4.
Men's top seed Novak Djokovic will end the eighth day of play later on Rod Laver Arena against Luxembourg's Gilles Muller.
