The Reds enjoyed the majority of the possession in the opening stages, but it could have been 1-1 within the first 15 minutes, as both sides squandered similar opportunities.
Salah quickly made amends, though, racing onto Coutinho's fine deep cross to steer a header inside the right-hand upright and beyond Schmeichel. Okazaki had the ball in the net after 39 minutes, and while the official called back the goal for offside, replays showed that should not have happened.
"He's always getting better because he can train all the time", said the Reds boss.
Individual mistakes at the back have undone Liverpool several times already this season, but their general lack of control was just as much to blame for Jurgen Klopp's charges nearly letting this match slip away from them.
Second half started with Leicester in the ascendancy - they were getting behind Liverpool's backline on repeat occassions.
Responding in kind, Klopp brought off Firmino and sent on Daniel Sturridge, with the striker providing a new dimension in the Liverpool forward line.
It looked like Liverpool were going to collapse completely as Vardy had the opportunity to then make it 3-3 from the penalty spot. He took his time after receiving a pass inside the box and fired it past Schmeichel.
The Reds goalkeeper Simon Mignolet was called into action as early as the fifth minute, denying Shinji Okazaki at close range following a break attack.
But Vardy smashed his penalty straight down the middle and Mignolet was strong to stop the ball.
"The boys deserved the three points".
Since the start of 2016-17, Liverpool have won more points from losing positions in Premier League matches than any other side (20) - however, they have also lost 20 points from winning positions in this period (5th most).