Ostapenko led 4-0 in the third set, then had a match point on Stakusic's serve while leading 5-4 and three more on her own serve while up 6-5.
Stakusic, who is ranked 155th in the world, won the last nine points of the three-hour, four-minute match to reach her first career WTA quarter-final.
Next up, Stakusic will oppose fifth-seeded Magdalena Frech of Poland, who defeated the United States' Ashlyn Krueger 3-6, 6-3, 6-1.
In earlier action, French No. 4 seed Caroline Garcia got past Japan's Ena Shibahara 7-6(3), 7-6(5).
Garcia ran out to a 4-1 lead in the first-set tiebreaker before finishing the job. Neither player broke serve in the second set, and Garcia slipped behind 4-3 in the ensuing tiebreaker before winning three straight points and eventually winning her second match point.
Garcia hit 10 aces and saved five of six break points in the one-hour, 50-minute battle.
Her quarter-final opponent will be sixth-seeded Marie Bouzkova of the Czech Republic, who handled Italy's Lucrezia Stefanini 6-2, 6-1.
Jasmin Open Tunisia
Eva Lys of Germany toppled No. 1 seed Elise Mertens of Belgium 1-6, 6-2, 7-6(4) in the round of 16 in Monastir, Tunisia.
Lys ran out to a 4-0 lead in the first set before Mertens won the next four games. Mertens briefly led the set 6-5, but Lys forced a tiebreaker that wound up swinging back and forth until Lys won the final three points to cement the upset.
Mertens out-aced her opponent 7-0 but also committed seven double faults and won a mere third (13 of 39) of her second-serve points.
No. 7 seed Lucia Bronzetti of Italy beat Ann Li of the US 6-4, 7-6(3), while Croatian Antonia Ruzic eased past British foe Yuriko Miyazaki 6-3, 6-1. Also, Turkey's Zeynep Sonmez saved 13 of 18 break points while posting a 6-4, 1-6, 6-3 victory over Belgian sixth seed Greet Minnen.