The home crowd thought Bolasie had scored in the 22nd minute when he curled one in from the edge of the box but despite being unsighted till the very last second, Karius managed to tip it around the post.
In addition, Nathaniel Clyne started a match for the first time in 11 months after recovering from a serious back problem, while central defender Ragnar Klavan was pressed into service at left-back, having played just 11 minutes since New Year's Day.
Liverpool's convincing victory over Manchester City in the Champions League was perhaps beyond Jurgen Klopp's wildest dreams, but has left him with an interesting dilemma going into Saturday's Merseyside derby at Everton.
Danny Ings and Dominic Solanke looked rusty as they were given rare starts, with the latter failing to take either of two early chances. It's easy to judge it, it was not Danny Ings" best game... that was the most importantly physical, the highest level he could play for ages and together with Dom [Solanke], they didn't play together often so it's not that they know exactly "when I go left, you go right' or whatever, that's all hard. Sadio was a threat, especially in the first half and did really well.
"When you miss those chances, it really is disappointing but, in the second half, we played much better, we took it to Liverpool and ran at them to create chance after chance". It was only during the last 10 minutes when Everton had a late surge, producing a number of opportunities which they failed to convert.
Liverpool goalkeeper Loris Karius also had to be at his best.
Tottenham would have displaced Liverpool if they had defeated Stoke with five goals or more. "If I keep a clean sheet, I am happy".
It was hardly a vintage derby though with Everton lacking any creativity and even the normal edge was missing from proceedings as the Reds extended their unbeaten record against the Blues to 17 matches stretching back to 2010.