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.
The 26-year-old Romanian showed no sign of the foot injury that forced her to withdraw from the Qatar Open last month and is well positioned to make a run at reclaiming the Indian Wells title she won in 2015.
Kasatkina improved to 11-6 on the WTA Tour season as she seeks to better her quarter-final appearance at Indian Wells 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.
"It was hard conditions to play in today because of the wind", she said.
"I just couldn't execute as well as I wanted to".
After recently taking back the world No. 1 ranking from Caroline Wozniacki, Halep is off to a fantastic start at the tournament.
That victory not only earned Wozniacki her first Grand Slam title but it made her the world number one.
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Simona Halep is into the quarterfinals at Indian Wells.
Wozniacki had a chance to take it back but she needed to win the Indian Wells title to do it.
Kasatkina next plays 10th seed Angelique Kerber, of Germany, who roared past seventh seeded Caroline Garcia 6-1, 6-1.
World number one Simona Halep did not need to be at her best as she beat Wang Qiang in straight sets to reach the quarter-finals of the Indian Wells Masters.
The World No.1 is now 17-1 on the season having already won a title in both singles and doubles at Shenzhen, China.
The first Croat to reach the round 16 at Indian Wells since 1997 when Iva Majoli lost to Venus Williams collecting her first top 10 win, beat 6-3 7-6 Marketa Vondrousova. The turning point against Wang, projected to move to the Top 55, came after a string of unforced errors left Halep trailing 5-4 in the first set. "So I will have to work hard", Halep told the Tennis Channel. "I had the crowd behind me and that felt good", said Williams. "But as I say, if you want to be on the top, you have to beat the top players".
If Wang was high on confidence following her performances earlier in the tournament, her morale was surely raised further as she twice responded immediately to falling a break down in set one.
Anisimova, a 16-year-old wild card, won her first three WTA Tour matches at Indian Wells, beating Pauline Parmentier, No. 23 seed Anastasia Pavlyuchenkova and two-time Wimbledon victor Petra Kvitova without dropping a set.