The US Open champion was beaten in just 57 minutes.
Raducanu, who had won her first two matches on the WTA Tour to reach the last eight in Cluj was beaten 6-2 6-1 by Ukraine’s Kostyuk, the sixth seed.
Third seed Raducanu lost her opening service game from 40-15 up and was broken five times in total as Kostyuk powered into the semi-finals in just 57 minutes.
Kostyuk, who will face top seed and home favourite Simona Halep in the last four, said in a courtside interview: “Honestly I don’t know how I won the first game.
"J'ai été 15-40 vers le bas, she had advantage, she was serving, and I think that’s where everything started. I somehow won that game and I gained this confidence.
“I had this feeling that I’m not going to lose this match and then after that I figured out by the fifth game maybe how to play her today – I don’t know, maybe next time I play her it’s going to be different tactic.
“But I figured it out and the tactic obviously helped me. I couldn’t read her because if her ball is in it’s really, really tough to do something. I was just trying to put as many balls back as I could.”
Raducanu made a total of 41 unforced errors and served seven double faults, two of them coming in what proved to be that crucial opening service game.
The 18-year-old did hold to love at her next attempt and created a break point in the sixth game, only for Kostyuk to save it with an ace and then break Raducanu again for a 5-2 conduire.
After comfortably serving out the set, Kostyuk broke in the opening game of the second and there was no way back for Raducanu, who at least showed her fighting spirit to save three match points before Kostyuk sealed victory.