Ankita, the world no. 293, fought tooth and nail against the higher ranked (265) Hesse but succumbed after a protracted battle that lasted 1 hour, 48 minutes with the French woman winning 6-3, 7-6 (4).
Hesse would meet top seed Aryna Sabalenka, ranked 96, in the semi-final following the latter's 6-4, 6-2 victory over the fifth-seeded Naomi Broady of Britain.
Earlier, Uzbekistan's unheralded Sabina Sharipova caused a major upset by rallying from a set behind to oust third seed Yanina Wickmayer of Belgium and enter the semis.
Jakupovic beat back the challenge of qualifier Alize Zim of France in straight sets, 6-3 6-1 in just under an hour in an earlier quarter-final at the Cricket Club of Indias courts.
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The match between Ankita and Hesse saw both the women hitting with power from the baseline.
Ankita had an early break in both sets but could not capitalise on it as she was broken back immediately.
With Ankita also managing to hold her serve in the next game before both women held serves again, tie breaker was enforced at 6-all in which Hesse took the lead with a mini break after 3-all.
The mini break was retrieved by Ankita but thereafter she fell behind 4-6 to give Hesse two match-points and the French woman finally clinched the set and the match when Ankita hit her forehand wide.
After winning the second set to restore parity, Sharipova traded breaks with her rival in the first two games of the decider before breaking her rivals serve in the third to go 2-1 up.
The 23-year-old Uzbek player had to save two break- points in the next game to hold her serve for a 3-1 lead and then broke Wickmeyers serve again to take a 5-2 lead.
She held her serve comfortably to come out a deserving winner against the 28-year-old Belgian who was a top-30 player for four years running between 2009 and 2012 and a semi- finalist at the US Open in 2009.
Results: Singles (Quarter finals): Dalila Jakupovic (SLO) bt Q-Alize Lim (FRA) 6-3 6-1; Sabina Sharipova (UZB) bt 3-Yanina Wickmayer (BEL) 3-6, 6-4, 6-2; Amandine Hesse (FRA) bt Ankita Raina (IND) 6-3, 7-6(4).
Doubles (quarter-finals): 2-Dalila Jakupovic (SLO)/Irina Khromacheva (RUS) bt Karman Thandi/Pranjala Yadlapalli (IND) 6-4, 6-2; 4-Julia Glushko (ISR)/Priscilla Hon (AUS) bt Junri Namigata (JPN)/Peangtarn Plipuech (THA) 7-6(5), 6-3; Q-Beatrice Gumulya (INA)/Ana Veselinovic (MNE) bt 1-Oksana Kalashnikova (GEO)/Veronika Kudermetova (RUS) 6-0, 6-1; Victoria Rodriguez (MEX)/Bibiane Schoofs (NED) bt Eva Wacanno (NED)/Yanina Wickmayer (BEL) 6-4, 7-6(4).