Both forwards scored for the first time this season, Minamino's double sandwiching a fine header from Origi, as Klopp's fringe men proved too good for the Canaries.
There was a scare for Liverpool at 1-0 when Norwich was awarded a penalty, but Caoimhin Kelleher saved from Christos Tzolis as the host squandered its best chance.
With Norwich bottom of the Premier League after five consecutive defeats, this was a chance to lift spirits, but it only served as a reminder of the squad's shortcomings.
Liverpool took a fourth-minute lead when Minamino turned sharply inside the six-yard box after Divock Origi headed down a corner from Tsimikas. The Japanese forward fired low past Angus Gunn for the swift breakthrough.
Ibrahima Konate powered a header just over the bar from another corner as Liverpool chased a quick second goal.
Kaide Gordon, a 16-year-old winger, showed promise for Liverpool and cut in from the left in the style of Mohamed Salah before flashing a left-footed shot just wide of the right post, while Curtis Jones shot just over from 18 yards.
Norwich lacked a punch up front and that was summed up by Adam Idah's tame side-footed shot that Kelleher smothered easily, and doubly so when the goalkeeper saved the spot-kick from Tzolis with his feet after 18-year-old debutant Conor Bradley fouled Dimitris Giannoulis.
Security of a second goal came in the 50th minute when Tsimikas played a lovely cross from the byline to Origi on the edge of the six-yard box, and the striker did the rest with a precise header into the right corner.
Origi went close again as he placed a shot just wide, before Gordon smashed the ball over from a presentable chance, 10 yards out, lacking the composure that will come with experience. Minamino showed him the way with 10 minutes remaining, slotting past Gunn after being set up by Alex Oxlade-Chamberlain.