Having raced into an early lead thanks to the talents of Philippe Coutinho, it looked like the visitors would have a fairly easy day, only for Leicester to show their infamous spirit and drag themselves back into the game not once, but twice.
However, the key moment came at 3-2, Mignolet saving a Jamie Vardy penalty to keep Liverpool's lead in-tact, securing three points.
Fast paced start
With Vardy shaking off injury to earn his place back in the team, Leicester had their favoured outlet available and used it to good effect early on.
Poor positioning from Dejan Lovren left Vardy free as he broke down the left, firing a shot at goal that was saved by Simon Mignolet, who had no control over Riyad Mahrez's rebound that, fortunately for Liverpool, went over the bar.
Salah should have opened the scoring with an open goal as Emre Can broke forward and hit a shot against the post, the ball flying back to the Egyptian who somehow stuffed his effort wide from just yards out though.
Salah and Coutinho net for Reds
He made up for it soon after though, nodding in the opener. It was fantastic work from Philippe Coutinho, linking up with Alberto Moreno before delivering an angled cross into the back-post, Salah emerging from the right to power a header past Kasper Schmeichel.
Coutinho would add a goal to his assist as Liverpool raced into a 2-0 lead, Moreno brought down by Wilfred Ndidi to give the Brazilian a chance, Coutinho making no mistake as he curled the ball beautifully into the top corner.
Anthony Taylor at the centre of controversy
Leicester would eventually get one back, but thought they'd done it a little quicker than they eventually would as Shinji Okazaki squeezed a shot in from a ball-in, Harry Maguire correctly flagged for offside as he'd won the flick on.
Okazaki would play a big part in the goal that Leicester grabbed in first-half stoppage time to halve Liverpool's lead, but not for particularly promising reasons.
Liverpool again failed to defend a corner, but may well feel aggrieved as replays showed Mignolet being tugged away by Okazaki as he looked to punch clear, Okazaki then breaking away from the Belgian to get on the end of a knock-down and divert the ball in to the net.
That goal rallied Leicester and the half-time break didn't seem to dampen their tempo, Liverpool struggling to control the game as the second-half kicked off.
Sturridge introduction aids Liverpool cause
Turning to one of his most experienced squad members, Klopp introduced Daniel Sturridge to good effect as he set up the Reds' third.
With Liverpool breaking at pace, it was Sturridge who showed the composure to slow things down and orchestrate, eventually sliding the ball across to captain Jordan Henderson who toyed with Schmeichel a little before slotting home.
Mignolet and Vardy battle ensues
Delighted and relieved, Klopp's mood quickly turned again, with Leicester bringing the gap back to one less than two minutes later.
The impressive Mark Albrighton got the better of Joe Gomez down the Leicester left, his ball in finding substitute Gray who had his shot saved by Mignolet, only for Vardy to nod in the rebound.
It looked like Liverpool were going to collapse completely as Vardy had the opportunity to then make it 3-3 from the penalty spot.
Andy King deserves credit for a fantastic through ball that set the striker away, but it was Mignolet's error that everyone took notice off as he missed the ball and clattered into Vardy, leaving Anthony Taylor with little choice other than to point to the spot.
With words exchanged between Vardy and his England colleague Henderson just before the penalty, it looked like Liverpool may have done their homework on the taker, Mignolet standing tall to beat away a powerful penalty that was struck straight down the middle.
That blunder didn't end the Leicester heat but certainly notched it down from boiling to a simmer, Liverpool resisting a late aerial onslaught for a first win in five.