Stakhovsky, who won last year's title as a lucky loser, was eliminated by Viktor Troicki of Serbia 6-3, 6-2 in the quarterfinals.
Troicki broke early in each set and again at 5-2 in the second. Stakhovsky, who had only three break points, had knocked out Troicki in the second round last year.
Troicki said he "waited the whole year'' to avenge that defeat against Ukrainian.
He will play second-seeded Cilic, Croatia's top player, on Saturday.
Cilic, who won the Chennai title last month, beat countryman Antonio Veic 6-2, 7-6 (6), and was impressed by the fight put up by the wild card, playing in his first ATP quarterfinals.
Veic led 2-0 in the second set, and rallied from 5-1 down to 6-6 in the tiebreaker.
"It was a hard match,'' Cilic said.
Seventh-seeded Ancic won another all-Croatian match against qualifier Ivan Dodig, 6-1, 6-3. Dodig had upset top-seeded Igor Andreev to appear in his first ATP quarters.
Next up for Ancic will be Jan Hernych after the Czech advanced when Mischa Zverev of Germany retired citing breathing problems while trailing 5-4.
Troicki beats 2008 champ Stakhovsky in Zagreb
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Troicki beats 2008 champ Stakhovsky in Zagreb
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