Salah put his side ahead from the penalty spot in the fourth minute after Robin Koch was harshly judged to have deliberately handled, but that strike sparked a flurry of goals that saw the sides share four of them in the next 20 first-half minutes.
Mateusz Klich's volley midway through the second half looked to have sealed a point for the newly-promoted visitors, but after substitute Rodrigo tripped Fabinho in the box, Salah again stepped up to slot home from 12 yards to seal three points and complete his hat trick.
If one thing can be taken away from a crazy game like this, it's that the Liverpool attacking engine is in no need of a jumpstart after the short off-season hiatus. Salah, in particular, looked like he'd never been away.
Defensively, the champions were all over the place at times, with Leeds pouring forward as they grew in belief that they could carve out chances. Against better teams, Liverpool could quite easily have lost today.
Manager rating out of 10
6 -- They got the job done, but Jurgen Klopp will be dragging his defence onto the training field on Monday to address a string of fundamental errors.
Player ratings (1-10; 10 = best, players introduced after 70 minutes get no rating)
GK Alisson, 6 -- Could do nothing about any of Leeds' goals, but the Liverpool stopper had to be alert on more than occasion to snuff out some promising openings for the visitors.
DF Trent Alexander-Arnold, 5 -- Left plenty of space in-behind as he ventured forward, and the England full-back should've done a lot better when trying to deal with Jack Harrison on the way to Leeds' first equaliser. Brought off in the closing stages and replaced by Joel Matip.
DF Joe Gomez, 5 -- Tried to help Alexander-Arnold deal with Harrison, but was unable to squeeze the ball away from the Leeds man as he wriggled away to score. Later on, got sucked towards the ball as Leeds played a looping pass in-behind on the way to their second.
DF Virgil Van Dijk, 6 -- The Dutchman rose highest to head home a corner to put his side 2-1 ahead, but the centre-back had a shocker on the half-hour when he skewed a casual clearance straight at Patrick Bamford for the forward to make it 2-2.
DF Andy Robertson, 6 -- An absolutely inch-perfect corner put the ball on a plate for Van Dijk to head home, and the Scot almost saw one of his crosses deflect in for a freak own goal soon after, Illan Meslier on hand to make a great save.
MF Jordan Henderson, 6 -- Hit a sweetly-struck volley straight down the throat of Meslier in the Leeds goal, in what was an otherwise low-key performance in a high-octane game. Brought off for Curtis Jones after 65 minutes.
MF Giorginio Wijnaldum, 5 -- Chugged away but didn't really offer the Liverpool midfield a great deal, and lost track of Klich as the Pole ghosted into the penalty area to fire home for 3-3.
MF Naby Keita, 6 -- Tried to drive the team forward with some piercing runs through the lines but faded in influence after a bright opening few minutes. Replaced by Fabinho just before the hour mark.
FW Mohamed Salah, 9 -- Emphatically lashed a fourth-minute penalty straight down the middle of the goal to put the Reds on their way, and then fired home brilliantly on the half-volley after 32 minutes to make the score 3-2. A cool, calm and collected spot kick late on sealed all three points for the Reds.
FW Roberto Firmino, 5 -- Declined to pull the trigger when in oceans of space in the penalty area early on, and the Brazilian looked particularly rusty on his first outing of the season. Picked up a yellow card after 75 minutes for pulling back Jack Harrison after losing possession in midfield.
FW Sadio Mane, 7 -- Didn't make it on to the scoresheet but Liverpool's No. 10 was still a menace in the final third and caused the Leeds defence all kinds of problems.
MF Fabinho, 6 -- Won his side an 87th-minute penalty to win it at the death.
MF Curtis Jones, 6 -- Should have done more to deny Klich an easy run into the box for Leeds' third goal.
DF Joel Matip, N/R -- Brought on to see the game out and he did just that.