Svitolina – the fourth seed – dropped the first set to 32nd seed Sorana Cirstea before fighting back to reach the round of 16 at the WTA Premier 1000 event on Sunday.
Two-time grand-slam champion Azarenka joined Svitolina in the next round after knocking out seventh seed Petra Kvitova, but 11th seed Halep became Aliaksandra Sasnovich's latest conquest.
Svitolina needed two hours, 32 minutes to outlast Cirstea 4-6 6-4 7-6 (7-3) and remain unbeaten in three career matches against the Romanian.
The Ukrainian was on the defensive throughout the opening set, facing 11 break points on her serve and managing to save eight of them before Cirstea finally won out.
Svitolina tightened things up from there, saving four of five break points the rest of the match.
"It was a very tough match today, and I was fighting and trying to find my game," said Svitolina, who will face Jessica Pegula next. "It was a bit of a rollercoaster.
"I wish I could play a little bit better in the first set, I had chances to grab that set, but unfortunately it didn't play the way I wanted. I had to fight for every point, and Sorana played a great match, I think. I'm happy that I could win today."
AZARENKA TAKES DOWN KVITOVA
Azarenka won her first two Tour-level matches against Kvitova in 2008 and 2009 but had prevailed only once in six meetings since then before Sunday's 7-5 6-4 triumph.
The two-time Indian Wells champion converted break-point chances when she needed them, six of 11 in all, and had 18 unforced errors to Kvitova's 26.
"It was important to just stay there, really take my opportunities, not to let her," Azarenka said after taking down two-time Wimbledon winner Kvitova. "If she gets in the groove in couple points, not to kind of let her extend that streak, if you want to call it that.
"I was trying to still create opportunities for myself, be more aggressive, and honestly just believing also that what I'm doing is right and see how I can execute that. So intention was good. Execution followed after."
After knocking off US Open champion Emma Raducanu in the second round, Sasnovich ushered out another grand slam winner in Halep 7-5 6-4 and will next face Azarenka.
The world number 100 had 22 winners to 16 for Halep and won 61.8 per cent of points on her serve as she reached the round of 16 at Indian Wells for the first time.
In other matches Sunday, ninth seed Anastasia Pavlyuchenkova fell to 23rd seed and US Open runner-up Leylah Fernandez 5-7 6-3 6-4, while second seed Iga Swiatek dropped only one game in a 6-1 6-0 demolition of Veronika Kudermetova.
Jelena Ostapenko and Shelby Rogers also advanced.