Three-times champion Williams stormed back from an early deficit to cruise past the British 11th seed, who needed several bouts of treatment on her lower back throughout a match that lasted two hours and 19 minutes.
Perhaps more impressive was that Williams, who turns 38 in June, was able to look so in control despite playing a day after spending nearly three hours on the court for her third round clash against Dutchwoman Kiki Bertens.
“The first set was definitely all her, and she came up with some amazing shots to break me at 5-all,” Williams told reporters.
“There was not much I could do there, except hope I could keep up in the second and third sets.”
Former world number one Azarenka, in only her second event after an eight-month absence because of a custody dispute involving her son, also looked sharp throughout her 6-2 6-2 win over 30th-seeded Pole Radwanska.
The Belarussian, competing as a wild card, won 79% of her first service points and needed only 82 minutes to prevail in a clash between former Miami champions.
Up next for the 28-year-old is Czech fifth seed Karolina Pliskova, who advanced to the last eight after Zarina Diyas of Kazakhstan was forced to retire injured while trailing 6-2 2-1.
“(Pliskova) is an amazing player, a top 10 player,” Azarenka said.
“For me, it’s a challenge to see how I can approach that and how I can progress from that match. It’s perfect. I wouldn’t ask for anything else.”
Former world number one Pliskova was efficient from the start and did not drop a point on her first serves in the entire match against Diyas, who took multiple medical time-outs and was visibly struggling with her movement.
Sloane Stephens raised her game to levels similar to last year when she won the US Open as she beat Spanish third seed Garbine Muguruza 6-3 6-4 and secure passage to the quarter-finals.
The American 13th seed dictated play against the Wimbledon champion from the start by taking eight of the first nine points of the match.
After securing a break in the fifth game of the second set, Stephens sent Muguruza packing with a backhand on the line on her second match point.
Stephens, who equalled her best result in Miami having also reached the quarter-finals in 2015, will next face German 10th seed Angelique Kerber, a 6-7(1) 7-6(5) 6-3 winner over Chinese qualifier Wang Yafan.
Among men, Alexander Zverev rallied from a one set deficit to defeat Spain’s David Ferrer 2-6 6-2 6-4 in the third round.
The 20-year-old German has yet to recapture the form that earned him five singles titles, including two Masters events, in 2017 but he has displayed guile in prevailing in a pair of three-set matches in Miami.
Zverev, ranked fifth in the world, recovered from a rusty opening set to break Ferrer’s service three times in the second set and twice more in the decisive stanza.
He will next face Australian Nick Kyrgios, who advanced with a 6-3 6-3 win over Fabio Fognini.
“I knew it was going to be tough. He has a lot of tricks up his sleeve and he’s very unpredictable,” Kyrgios said.
“We both competed. I just played a little better in some bigger points.”
In earlier action, Fernando Verdasco dug deep to earn a 3-6 6-4 7-6(4) win over Australian qualifier Thanasi Kokkinakis to set up a fourth round clash against fellow Spaniard Pablo Carreno Busta.
Also reaching the fourth round were Kevin Anderson, a 4-6 6-2 6-3 winner over Karen Khachanov, and Denis Shapovalov, who upset 11th-seeded American Sam Querrey 6-4 3-6 7-5.
Nadal back for Spain in Davis Cup squad
Madrid: Rafael Nadal celebrates his impending return to world number one spot with a recall to Spain’s Davis Cup team for the quarter-final tie against Germany.
Nadal, who will top the latest world rankings next week rejoins the squad for the first time since 2016 for the April 6-8 tie, team captain Sergi Bruguera announced.
The 16-time Grand Slam winner is certain to take over from Roger Federer as world number one after the shock defeat of the Swiss ace in the second round of the Miami Masters on Saturday.
Bruguera said that the 31-year-old was recovering well from an injury to his right leg that has sidelined him since he pulled out of the Australian Open in January. In February, he pulled out of the Mexican Open just hours before his opening match.
“I have kept in touch with him and he is going from strength to strength and he is feeling good after his convalescence and he is training more,” said Bruguera.
Nadal last appeared for Spain in the Davis Cup in September 2016.
Pablo Carreno, Roberto Bautista, David Ferrer and Feliciano Lopez line up with Nadal for the tie on clay in Valencia, eastern Spain.