Halep breezed through her fourth round match with a 7-5, 6-1 win over China's Wang Qiang and eighth seed Venus Williams faced a stern test from Anastasija Sevastova but held on for a 7-6 (8/6), 6-4 victory.
The 20-year-old Kasatkina, who has beaten all four reigning Grand Slam champions in the past year, continued her giant-killing form with a 6-4, 7-5 shock victory over world No. 2 Wozniacki.
Kasatkina recorded her second straight win over Wozniacki, having also beaten the Australian Open champ at the St. Petersburg tournament last month.
"I'm playing my best matches against the best players," she said. "But if you want to be on the top, you have to beat the top players. So quite a simple rule."
Kasatkina improved to 11-6 on the WTA Tour season as she seeks to better her quarter-final appearance here in 2016.
Kasatkina hit one ace, won 54 percent of her first serves and broke Wozniacki's serve six times in the one hour, 40-minute match.
"She outsmarted me today," Wozniacki said. "But, you know, at least I tried, and I tried to do what I thought I needed to do out there today.
"I just couldn't execute as well as I wanted to."
It was a disappointing exit for Wozniacki, who finally silenced her critics earlier this year with her maiden Grand Slam title in Melbourne, where she defeated Halep in the
final. Wozniacki had a chance to get the No. 1 back but she needed to win the Indian Wells title to do it.
Kasatkina said she is starting to believe in herself, especially when she faces top 10 players.
In the beginning it was really tough because I was really nervous going on the big stages against big players," she said.
"But now with experience and with time I'm getting into it.
"I'm getting pretty confident at this level."
But she added, "I am still young so I have to work on everything."
Kasatkina next plays 10th seed Angelique Kerber, of Germany, who roared past seventh seeded Caroline Garcia 6-1, 6.
Meanwhile, Halep, who is one of two former champions left in the draw, is now 17-1 on the season having already won a title in both singles and doubles at Shenzhen, China.
Halep advances to the quarter-finals where she will face unseeded Petra Martic of Croatia who beat Czech Marketa Vondrousova 6-3, 7-6 (7/4).
Halep called for a timeout in the first set so her coach Darren Cahill could come down to the court to discuss strategy.
"It was really important, because he just told me what I have to do. I knew that I had to change something, but I didn't realize what it was exactly," she said.
"So it was a good chat. I did exactly what he said after that."