"I'm excited to play my good friend in the semi-finals,” she said. “We've already played in the US Open final and it's really good for American tennis.”
Kasatkina wore down the second seed Caroline Wozniacki in the last 16 but Stephens proved too solid for the Russian.
The pair traded breaks early in the match, before Stephens produced some ferocious hitting at the end of a marathon eighth game to give herself the chance to serve for the opening set.
She claimed it with the help of a sumptuous backhand drop shot.
Stephens broke in the third game to lead 2-1 in the second set and ran away with the tie to advance after 70 minutes.