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Third seed Coco Gauff served up 17 double faults but still managed to beat China's Yuan Yue 6-4, 6-3 to reach the women's semifinals in the Californian desert for the first time.
The top remaining seed after Saturday's early exit for world No. 1 Iga Swiatek, Sabalenka's biggest enemy has often been her own emotions but she continued to keep them strictly under control to reach the last eight without dropping a set.
Russian teenager Mirra Andreeva produced a thrilling comeback from 5-1 down in the third set to beat Frenchwoman Diane Parry 1-6, 6-1, 7-6(5).
Fourth seeds Coco Gauff and Jannik Sinner, along with home hope Alex de Minaur and world number 11 Barbora Krejcikova progressed.
He will be replaced in the draw by Zizou Bergs.
Top-ranked Iga Swiatek faces a tricky test against 2020 champion Sofia Kenin first up in the women's section.
Home favourite Sinner also clinched top spot in the group with a 3-0 record.
The earlier Green Group clash was a non-event as Stefanos Tsitsipas retired injured against Holger Rune.
Rublev, ranked fifth in the world, began to despair after losing the first set and failed to put up much of a fight in the second.
Jannik Sinner made a dream start in front of his home crowd with a 6-4, 6-4 defeat of Stefanos Tsitsipas in their opening Green Group clash.