Goalkeeper Swaleha became the star with two saves to guide Delhi into the quarterfinals after the match ended in a 2-2 draw following the regulation time and 15-minute extra time.
Delhi scored a goal in the 16th minute through Lalita but Rajasthan managed to level the scores in the 59th minute with the penalty corner conversion by skipper Sonam.
Delhi once again took the lead in the 65th minute via a penalty stroke by Lalita but three minutes later Sonam yet again scored the equaliser for Rajasthan.
The match's result was ultimately decided by penalty shoot-out.
In another match, Sabnam Lakra scored four goals to help last year's runners-up Jharkhand register a 9-0 win against Manipur in its opening Pool B game.
Jharkhand dominated the proceedings from the start. Besides Sabnam Lakra, other goal scorers were Sushila Lakra, Anima Minz, Punam M Tete, Nomita Xalxo and Ranjita Soreng.
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In Pool D, Odisha blanked Puducherry 29-0. Surabhi Pradhan (5), Punam Barla (4), Reshma Lakra (4), Lilima Minz (4), Biranji Ekka (3), Anita Ekka (2), Rajkumari Ekka (2) and Lata Kujur (2) scored for the winning team.
In another Pool D match, Karnataka beat Gujarat 18-0. Uttar Pradesh made a good start in Pool F as they defeated Jammu & Kashmir 19-0.
After taking time to settle down, Uttar Pradesh scored first in the 11th minute through Archana Singh and thereafter scored through Pratibha Chaudhary, Kavita Maurya, Kiran Kumari, Komal Sahani, Srishti Singh, Pranjal Sharma and Vaishali Lal.