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Norwich grew into the game as the half wore on, putting pressure on a rotated Liverpool back line before seeing visiting keeper Caoimhin Kelleher deny Christos Tzolis from the penalty spot ahead of the break.
Origi doubled Liverpool's lead shortly after the restart, redirecting a Konstantinos Tsimikas cross past Gunn with lovely a glancing header that found its way into the side netting.
Minamino put the match away with his second in the 80th minute, sliding into the area, finding space and poking home to put Liverpool three goals to the good and book passage to the fourth round of the annual tournament.