The reigning Australian Open champion beat World No. 130 Dodin 6-1, 6-7(5), 6-2 to set up a semifinal clash against Belgium's Alison Van Uytvanck, who defeated France's Caroline Garcia 6-2, 6-2 later on Friday.
No. 1 seed Kenin raced to a one-set lead in the opening set, breaking Dodin's serve three times, before losing in a second-set tiebreaker. The American collapsed from a 5-1 lead in the tiebreaker to lose the set 7-6(5).
Dodin kept her hopes for an upset alive in the third set when she broke Kenin's serve to level the match again. But the American quickly dispatched Dodin after that, easing the remainder of the match to reach the semifinals.
Kenin, 21, is chasing her fifth career WTA-singles victory -- it would be her first since winning her first Grand Slam in Melbourne in January.
Kasatkina defeated Italy's Camila Giorgi 6-2, 4-6, 6-2, to book her place in the final four.