At a set and a break up, the British No 1, 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 who enjoyed a 6-2, 6-3 quarter-final win over Anastasia Pavlyuchenkova
Kasatkina, ranked 16 places below Konta at 26 in the world, claimed an upset win over world No 1 Angelique Kerber on Tuesday but was left to rue making double the amount of unforced errors (30) as winners (15).
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.
Former world No 1 Caroline Wozniacki’s streak of not advancing past the quarter-finals in her past seven Sydney Internationals was extended.
Denmark’s Wozniacki lost 7-5, 6-7 (6), 6-4 to Barbora Strycova in a duel that stretched to three hours, 19 minutes.
Both players were treated for foot injuries during a grueling second set at Ken Rosewall Arena. Wozniacki came back from 5-2 down in that set, and then 5-0 in the tiebreaker, to win it and force a third set.
“It was brutal out there ... but you just try and think like you’re on a beach drinking pina coladas,” Wozniacki said. “That’s basically your train of thought. You know that it’s the same for both players, so I was just trying to mentally just try and keep cool.”
In the semi-finals, Strycova will play Agnieszka Radwanska. The second-seeded Pole beat Chinese qualifier Duan Yingying 6-3, 6-2.
At the Hobart International, former French Open finalist Lucie Safarova was beaten 2-6, 6-3, 7-5 by Japanese qualifier Risa Ozaki. Ozaki will next meet Monica Niculescu.
Top-seeded Kiki Bertens advanced with a 6-1, 6-4 win over Galina Voskoboeva of Kazakhstan and Shelby Rogers defeated Lara Arruabarrena of Spain 6-2, 6-2.