Minamino, one of 10 changes to the Liverpool team that beat Chelsea on penalties in Sunday's League Cup final, struck the opening goal in the 27th minute after Divock Origi laid a left-wing cross into his path.
The former Red Bull Salzburg attacker doubled the hosts' advantage in similar fashion six minutes before the half-time interval. This time he slammed a shot home at the back post after a corner dropped into his path.
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Norwich, bottom of the Premier League and facing an uphill task to avoid relegation, put the result in doubt when Lukas Rupp fired home an unstoppable strike from outside the box.
But despite Jurgen Klopp continuing to rotate his lineup in the second half amid what promises to be a hectic climax to the season, Liverpool had enough to see out the victory.
"We have never been in an FA Cup quarterfinal [with Klopp in charge]," Klopp told ITV. "We made 10 changes to the team and we were always going to find it tough against a good team, but we played really good football and scored some really good goals.
"Takumi is an incredible player and tonight was the best game he had for us. He was a constant threat throughout."
While Klopp has played down such talk. Wednesday's win means Liverpool's hopes of a historic quadruple remain alive. After attempting to close the gap on Premier League leaders Manchester City when they host West Ham on Saturday, Liverpool will take a 2-0 lead into the second leg of their Champions League Round of 16 tie with Inter Milan three days later.
Information from Reuters was used in this report.