Japan international Minamino made it five goals in four EFL Cup appearances before Origi scored for the 11th time in his 10th start in the competition with an impressively improvised back heel volley.
Having been fortunate not to go behind in the first half when Sean Maguire, Ryan Ledson and Brad Potts failed to take their chances, Minamino made the breakthrough for Liverpool in the 62nd minute.
The Premier League giant dominated possession throughout and were sure of a place in the quarter-finals for just the second time in five seasons after Origi somehow flicked the ball home with six minutes remaining.
Preston initially looked more dangerous despite seeing less of the ball and ought to have taken the lead in the 28th minute when they had three chances in rapid succession.
Adrian did brilliantly to deny Maguire from six yards after Joe Gomez was dispossessed, and Neco Williams's goal-line clearance kept Ledson out on the follow-up. The rebound fell to Potts, but he sliced his attempt high and wide.
After wasting a chance at the start of the second half, there was no mistake from Minamino with his next opportunity as he stabbed in a cutback from Williams after a brilliant pass from Tyler Morton released the right-back.
Kostas Tsimikas then rattled the crossbar with a deep cross and the ball fell to Williams, whose blocked shot sat up nicely for Origi to steer in a spectacular late second.