The Reds have endured a miserable recent run and their FA Cup victory over fourth-tier Plymouth Argyle remains their only win this year.
Even the return of the influential Sadio Mane to the starting line-up failed to inspire a revival, with Simon Mignolet again at fault for debutant Alfred N'Diaye's opener on the stroke of half-time.
The visitors, who did not register a shot on target in the first half, carried a greater threat after the interval, but saw Philippe Coutinho squander the best of their chances.
Having recorded a 5-1 win over Hull in September, the contrast in the fortunes of Klopp's side then and now was underlined by Niasse's 84th-minute goal – his first in the Premier League.
For the Tigers, revitalised under Marco Silva, a fourth consecutive home win in all competitions has breathed new life into their flourishing survival bid.
The Portuguese had one alteration forced upon him as Michael Dawson limped out of the warm-up, with Andrea Ranocchia handed a full Premier League debut.
N'Diaye had an early sight of goal but fluffed his lines to shoot tamely at Mignolet from the edge of the box.
Fellow loanee Omar Elabdellaoui required treatment for a nasty facial injury after an unintentional elbow from James Milner, but returned apparently untroubled by the blow.
The hosts escaped unscathed from a moment of danger in the 21st minute, with Eldin Jakupovic and Harry Maguire getting in each other's way and the loose ball falling to Coutinho, whose effort deflected wide.
And on the stroke of half-time Silva's side had the lead, N'Diaye presented with the simplest of finishes after Mignolet failed to deal with Maguire's header.
Jakupovic was finally tested soon after the restart and initially fumbled Emre Can's diving header, but gathered at the second attempt.
He was much more impressive in keeping out Mane's looping header as the visitors cranked up the pressure, though Abel Hernandez raced through at the other end and may have doubled the lead but for a heavy touch that favoured Mignolet.
Hull were almost made to pay for that in the 66th minute when Coutinho was left with a free strike six yards out, but made a poor connection to slash well wide.
Klopp made two late changes as Divock Origi and Alberto Moreno were thrown into the mix, but within minutes Mane conceded possession high up the pitch to prompt a swift Tigers counter that ended with Niasse coolly slotting in the second.
Jordan Henderson's strike forced a stunning last-minute save from Jakupovic, denying Liverpool even a consolation on another miserable afternoon for the Merseysiders, whose title bid is now surely over.