Dart was the last remaining British player in the women's tournament following Emma Raducanu's early exit.
The 25-year-old beat Olympic bronze medallist Elina Svitolina on her way to the fourth round. As a result of her run, Dart will break into the WTA's top 100.
The 2021 Wimbledon mixed doubles finalist will also become the new British number two after overtaking Heather Watson.
Australian Open semi-finalist Keys proved a match too far as the 25th seed progressed 6-1 6-4 in just over an hour to reach the quarter-finals in Indian Wells for the first time.
Speaking to the Tennis Channel, the American talked through her breakthrough in the second half of the match, saying: "I just kept thinking I had so many opportunities to break even in the second set, and I felt I was holding my serve pretty comfortably.
"I just kept telling myself to keep giving myself these opportunities and put myself in that position to finally get that break. When I was able to [make it] 5-4, a weight stood off my shoulders."
Keys also underlined the importance of keeping her cool despite the slow speed of the courts, adding: "It's an acceptance that point are probably going to be longer and I'm not [getting] irrationally angry that the courts are slow.
"Just accepting that it is what it is and not trying to not only fight my opponent but also the surface, because I'm going to lose that battle."
Keys goes on to meet number three seed Iga Swiatek, the 2020 French Open champion, in the last eight after the Pole came from behind to defeat Angelique Kerber 4-6 6-2 6-3.
Simona Halep defeated fellow Romanian Sorana Cirstea 6-1 6-4 and goes on to play Petra Martic for a place in the semi-finals.