The Canaries had a great chance to equalise at the start of the second half as Giannoulis won the ball back on the left and played a low cross into the area for Lees-Melou to fire over from 10 yards out.
Liverpool then made it 2-0 on 50 minutes as Konstantinos Tsimikas, overlapping down the left, curled a cross to the edge of the six yard box for Divock Origi to flick a header into the bottom corner.
And the Belgian could have had a second on 53 minutes as he latched onto a pass from debutant Kaide Gordon but he sent a right foot shot just wide.
Farke then made a double change, with just over 20 minutes to go, as Teemu Pukki and Milot Rashica came on for Mumba and Tzolis.
With 16 minutes to go Andy Roberston gave away possession cheaply to Rupp but his effort was wide of the right hand post.
And then Minamino made sure of the victory with 10 minutes to go as Alex Oxlade-Chamberlain won the ball back high and feed the Japanese strike who fires under Gunn to make it 3-0.
*Norwich City: Gunn, Mumba (Rashica, 69), Omobamidele, Hanley, Gibson, Giannoulis; Lees-Melou (Dowell, 77), Gilmour, Rupp; Tzolis (Pukki, 68), Idah.
*Subs not used: Krul, Kabak, Sørensen, Williams
*Bookings: Lees-Melou (70), Rupp (90)
*Liverpool: Kelleher, Bradley, Konaté, Gomez, Tsimikas (Robertson, 66), Keïta (Morton, 46), Jones (Henderson, 86), Oxlade-Chamberlain, Gordon, Minamino, Origi.
*Subs not used: Adrian, Jota, Phillips, Morton, Balagizi.
*Goals: Minamino (4), Origi (50), Minamino (80)
*Venue: Carrow Road
*Tonight’s official Golden Goal times were:
1st Goal: 03:29
2nd Goal: 49:43
3rd Goal: 79:12
Tonight’s lucky number was 0410 – the winner will get a signed Norwich City shirt. The Golden Goal tickets for tonight were issued as ‘OPPONENT B’