Virgil van Dijk celebrates scoring Liverpool's fourth goal in an FA Cup tie at home to Norwich
Klopp, who revealed his impending departure on Friday, was given a rousing reception by the home supporters before kick-off, and his team wasted little time in moving through the gears against the Championship play-off hopefuls.
Darwin Nunez had already struck the post with a magnificent curling drive by the time Curtis Jones nodded the hosts in front from full debutant James McConnell's cross.
The Canaries equalised against the run of play when Ben Gibson steered in Gabriel Sara's corner, but Nunez restored Liverpool's lead with a calm finish after fine work from full-back Conor Bradley.
And it was 3-1 early in the second half when Diogo Jota latched on to a wayward header from Gibson before thumping a low shot past goalkeeper George Long.
Second-half substitute Virgil van Dijk, left completely unmarked from a Dominik Szoboszlai corner, headed home Liverpool's fourth before a superb Borja Sainz strike gave Norwich hope - moments after Christian Fassnacht's finish was disallowed for offside.
The only other sour note for the home side was a late injury for teenager McConnell, who had to be replaced by Luis Diaz with 12 minutes remaining - but there was still time for Ryan Gravenberch to complete the scoring with a close-range header.
The Reds will host Watford or Southampton in the fifth round after the Championship clubs played out a 1-1 draw at Vicarage Road earlier on Sunday.
-- Courtesy of BBC Sport