Klopp had urged his players not to waste that dramatic win over his former side two weeks ago and Joe Allen, on his first European start since 26 November and only his fourth in the competition this season, almost got them off to a flier.
Arriving into the penalty area at speed in the fifth minute he could only direct Adam Lallana's cross straight at Sergio Assenjo when a shot a yard either side of goalkeeper would have produced the breakthrough.
Without the focal point of Divock Origi, who injured ankle ligaments a week ago in the Merseyside derby but fortunately does not require surgery and could make a return before the end of the season, Liverpool were aimless in the final third.
Too often crosses were swung in at no one in particular, giving centre-backs Eric Bailly and Victor Ruiz an easy ride.
As a result Villarreal had the better chances with former Tottenham Hotspur striker Roberto Soldado off target with two shots at either end of the half, while Simon Mignolet got down well to parry Tomas Pina's low effort.
Villarreal right-back Mario Gaspar also had efforts snuffed out by Kolo Toure, continuing in central defence with Mamadou Sakho now officially suspended by UEFA for a failed drugs test - and Lovren.
The ineffectual Philippe Coutinho - reportedly being watched by Barcelona scouts - was replaced by winger Jordon Ibe at the break after feeling unwell but within 40 seconds of the restart Cedric Bakambu headed against the post from Jonathan dos Santos's corner.
Firmino, having kept himself onside, responded midway through the half with a shot which Asenjo tipped onto the upright - but that was as close as the visitors came.
Mignolet came to their rescue three minutes from time when Bakambu seized on Ibe's mistake while charging fullback Alberto Moreno ignored the unmarked winger from a quick counter-attack and instead fired into the side-netting.
But in added time substitute Adrian slid home after Denis Suarez had outpaced Toure.
