At a set and a break up, the British number one, seeded sixth here, appeared to be coasting towards a last four place but her Russian opponent broke back immediately and then again to establish a 4-2 lead.
But Konta, who made 31 winners and 30 unforced errors, dug deep to win five of the next six games for a 6-3 7-5 victory in just under an hour and three-quarters as she set up a showdown with Canadian wildcard Eugenie Bouchard.
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Kasatkina, ranked 16 places below Konta at 26 in the world, claimed an upset win over world number one Angelique Kerber on Tuesday but was left to rue making double the amount of unforced errors (30) as winners (15).

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The 19-year-old made a sluggish start in searing temperatures by dropping her opening service game and although she got back on serve, Konta claimed what would prove to be a decisive break in the fourth game.
Konta broke in the first game of the second set but then dropped serve twice before hitting back immediately to level matters at 4-4.
The Sydney-born 25-year-old continued her resurgence by breaking Kasatkina for the final time in the 11th game and then held serve to complete a hard-fought triumph.
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