Barcelona, resting several key players including Lionel Messi, Xavi and Andres Iniesta, created a host of chances at a rain-drenched Municipal de Foietes stadium in Benidorm but were unable to break the deadlock until the second half.
The 18-year-old Krkic, who fired wide with only the goalkeeper to beat in the first half, picked the ball up in midfield in the 54th minute, ran at the defence and curled a measured shot into the right-hand corner.
Barcelona have the most King's Cup triumphs with 24 but have not lifted the trophy since 1998. Last season, Valencia beat Getafe 3-1 in the final to claim their seventh Cup.
In Wednesday's ties, the lowest-ranked side left in the competition, Portugalete of the Tercera division (fourth tier), play La Liga leaders Valencia.
The arrival of the top-flight clubs in the tournament also sees third-tier Real Union play league champions and 17-times winners Real Madrid on Thursday. Return matches are scheduled for November 12.