Minamino broke the stalemate in the 62nd minute on he slotted in Neco Williams' clip across the box, with Origi doubling the lead late in the 84th with highlight-reel backheel effort.
Jurgen Klopp's side withstood a lacklustre first half at Deepdale that saw keeper Adrian keep the Championship hosts at bay and keep it scoreless going into the break.
The home side's finishing in front of goal turned costly as Minamino guided Williams's cross into the back of the net, allow the Reds to settle down and enjoy great spells of possession.
The game was sealed in the 84th minute when Kostas Tsimikas put in a hopeful cross that deflected off of the bar, allowing Williams again to divert the ball with Origi spectacularly flicking the it over Preston netminder Declan Rudd.
"We didn't play well, that's how it is. We started quite OK, but then we lost the structure completely and that was the problem," Klopp said. "We were too lively, too desperate to get the ball.
"As a team we can play better, but individually I saw a few really nice performances. It's all about getting through to the next round, that's what we did. That it [would be] difficult with 11 changes, that was clear."
Preston manager Frankie McAvoy said: "We made nine changes [from our previous match] but there's not a lot between the players in our squad, it's fine margins. Sometimes you get it wrong but tonight I felt the guys who were picked did us proud."