The visitors led after four minutes as Minamino produced a sharp turn and finish from Divock Origi’s knockdown.
Caoimhin Kelleher denied Norwich an equaliser before half-time, saving Christos Tzolis’ penalty with his foot.
Origi headed in Konstantinos Tsimikas‘ cross five minutes after the break and Minamino slotted in a late third.
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Both managers named nine changes from the weekend’s Premier League matches,
with Liverpool boss Jurgen Klopp handing debuts to teenagers Kaide Gordon and Conor Bradley from the start – plus Tyler Morton as a substitute.
The highly rated 16-year-old Gordon, signed from Championship side Derby County.
For an initial £1.1m in February, hinted at his potential as he fired narrowly wide after driving inside from the right midway through the first half.
Norwich improved as the opening period wore on but Adam Idah shot directly at,
Kelleher and Tzolis failed to deliver from the penalty spot after Bradley had fouled Dimitris Giannoulis.
And Liverpool moved out of reach after Pierre Lees-Melou missed a glorious opportunity to equalise,
early in the second half, with Minamino’s well-taken second from Alex Oxlade-Chamberlain’s through ball sealing a comfortable victory.
