The Uruguayan has now scored 11 in his last four appearances against Norwich, with this his third treble in that time, and three of the strikes were of the highest quality.
The opener came after quarter of an hour as he thumped in a long-distance volley, before turning home a corner 14 minutes later.
Liverpool made three changes from the side that lost to Hull City, while Norwich had just one alteration from the weekend as Johnson replaced Gary Hooper.
It was the home side who started on the front foot and Suarez gave his side the lead in spectacular fashion with 15 minutes gone.
John Ruddy made poor contact with a goal-kick and, as the ball bounced in favour of the Uruguayan, he looped a first-time volley high over the goalkeeper and into the corner from 40 yards.
Suarez doubled the advantage in the 29th minute, hooking home Philippe Coutinho's left-wing corner after being given too much space six yards from goal, and six minutes later, the striker completed his treble with another stunning strike.
After receiving the ball from Sterling, he drove forward, chipped the ball around Leroy Fer and sent a searing half-volley beyond Ruddy from 20 yards to cap an exhilarating first-half performance.
Sebastian Bassong sent a free header over the crossbar four minutes after the break but two minutes later it almost got worse for Norwich as Ryan Bennett had to clear Steven Gerrard's goalbound header off the line.
Joe Allen was the next to threaten the visitors' goal, bringing a flying save out of Ruddy with a thumping effort from 25 yards before the England international dived low to his left to deny Suarez.
In a frantic finish, Johnson flicked home Nathan Redmond's cross eight minutes from time, Gerrard hit the post and Sterling converted Suarez's pass as Liverpool bounced back from Sunday's disappointing defeat to Hull City in style.
