Klopp opted for a change of his own on 68 as Alex Oxlade-Chamberlain for James Milner as the pushed for the opening goal.
The 0-0 draw may hurt Liverpool's chances of finishing second in the Premier League, but forgive the majority of Anfield for not caring too much given an explosive and comprehensive 3-0 win over Manchester City in the first leg of their UEFA Champions League quarterfinal on Wednesday.
At the other end Karius was called into action to quickly move across his goalline and tip a Bolasie curler away with his fingertips.
After a hard start to life at Liverpool, Karius has managed to come into his own in the past couple of months and has successfully dislodged Simon Mignolet from the starting spot between the sticks.
Wayne Rooney looked unhappy to be substituted for the fit-again Idrissa Gueye early in the second half but the impetus remained with Liverpool, who brought on Alex Oxlade-Chamberlain and Firmino, until a frantic final 10 minutes in which Seamus Coleman and Dominic Calvert-Lewin went close to grabbing an unlikely victor for the home side.
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Both goalkeepers were predictably happy with not conceding in a game in which both delivered assured performances. "So I'm not really over the moon".
At the end of the month we'll go through the results of the month's polls and vote The Liverpool Offside's player of the month and, come next May, those monthly results will help to determine your TLO player of the season for 2016-17.
Dominic Solanke, starting down the middle with Roberto Firmino on the bench, was the main culprit as he wasted the two clearest chances of a drab encounter in front of England manager Gareth Southgate, while Loris Karius only had one Everton effort to deal with across the entire 90 minutes. "It was tough conditions for us keepers". He kept another clean sheet which now puts him 13 clean sheets in 22 games as well as 8 in his last 12.
That is the opinion of former Everton midfielder Lee Carsley. "I'm really angry about a few people who make decisions in this league".
Liverpool remained on third place as Tottenham Hotspur defeated Stoke City 2-1.