Seeded second, the Greek player was 3-1 down in the third before breaking back and beating Khachanov 4-6, 6-3, 7-5. Tsitsipas is seeking his first tour title since February 2020.
His win set up a semifinal against fourth-seeded Andrey Rublev, who was pushed hard by qualifier Jeremy Chardy in a 7-6 (2), 6-7 (2), 6-4 win.
Chardy saved two match points at 6-5 down in the second set. In the decider, Rublev sealed the win with his 15th ace on his fourth match point. Rublev is aiming to reach his first final of 2021 after going 5-0 in tour finals last year.
In the other half of the draw, the semifinal pits Borna Coric against qualifier Marton Fucsovics.
Coric hit 11 aces on his way to a hotly contested 7-6 (2), 7-6 (4) win over Kei Nishikori in their quarterfinal.
Fucsovics won 6-4, 6-3 over Tommy Paul.