Jacob Jonathan gave the People’s Elephant the lead in the 25th minute before Aderemi Adeoye equalised eight minutes from full-time.
Ikorodu City converted all five spot-kicks, while Enyimba managed four to bow out.
Elsewhere, El-Kanemi Warriors edged Bendel Insurance 8-7 on penalties, while Nasarawa United defeated Rivers United 5-3 in another shootout after a 2-2 draw.
Barau FC beat Sunshine Stars U-19 2-1, Plateau United overcame FC Basira 2-1, and Wikki Tourists sealed a 3-1 victory over Kada Warriors to book their places in the quarter-finals.
Results
Sokoto United 2-0 Adamawa United
Bendel Insurance 3-3 Elkanemi Warriors (7-8 on pen)
Kada Warriors 1-3 Wikki Tourists
Bida Lions 0-1 Flight FC
Rivers United 2-2 Nasarawa United (3-5 on pen)
Plateau United 2-1 FC Basira
Ikorodu City 1-1 Enyimba FC (5-4 on pen)
Sunshine Stars U19 1-2 Barau FC