The No.13-seeded Stephens was the first of the two to advance, winning six games in a row from 5-3 down in the first set to take complete control against Elina Svitolina and eventually prevailing, 75 64.
"I was playing pretty well at the beginning, but it was definitely tricky because the wind was blowing really hard on one side and not really on the other side," Stephens said after the match. "It was a little bit rough out there, but after I got my rhythm and found what I wanted to do, it was easier."
Azarenka, the No.2 seed and two-time defending champion, traded breaks in the first two games of her match against Yvonne Meusburger but then won 11 games in a row to cruise past the Austrian, 61 60.
"I felt that I was doing what I wanted to do," Azarenka said. "I think the beginning had quite a few long rallies, but I took advantage of that and raised my level. I'm quite happy with that. It was just a good performance, I think. And that I could maintain that throughout the match, that was important for me."
Azarenka beat Stephens in the pair's only previous encounter, in the semifinals here a year ago.
"It was a tough match - obviously semis of a Grand Slam, it was pretty intense," Stephens said. "But last year has nothing to do with this year. It's a totally different year. A lot of things have happened.
"I've been playing well and I'm just looking forward to getting back on the court."
"Sloane is a great player and a very tough competitor," Azarenka said. "She has a great record here. It's going to be tough and every round is more challenging now. I'm looking forward to that match.
"I also feel like she's improved so much. She proved that she can handle big stages. She can play against top players. So it's always tough when a player like that knows how to handle themselves."
Another fourth round match-up that was set up was No.8 seed Jelena Jankovic against No.11 seed Simona Halep. Jankovic rallied from 5-2 down in the second set - and saved a set point down 5-4 - to beat Kurumi Nara, 64 75, while No.11 seed Halep beat Kazakh qualifier Zarina Diyas, 61 64.
Halep leads Jankovic in their head-to-head series, 2-1, though all three matches have been extremely close, as evidenced by the fact that Jankovic held match points in their last two but lost both.