Salah gave the Reds the lead with a penalty early in the game, but Jack Harrison fired Leeds straight back on level terms, with a superb goal.
Virgil van Dijk scored a free header a few minutes later, but the Reds were again immediately pegged back by Patrick Bamford, after an awful mistake from van Dijk himself.
Salah blasted Liverpool back ahead two minutes after that, and that's how it stayed until midway through the second half, when a superb build-up ended with Mateusz Klich putting Leeds back level for a third time.
Salah, though, had the final say after Liverpool were gifted a penalty by a daft Rodrigo Moreno challenge on Fabinho in the box.
The champions knows that they will eventually have to produce better performances in the defence of their crown, but they will definitely take these three points on what was an awfully difficult opening day fixture.
Here's how each player fared for both sides in this game.
Liverpool Player Ratings
Alisson Becker - 5/10
He probably might have done better with guarding his near post for the first goal, but that was about the only thing he can be faulted for. He was left exposed by some shambolic defending in front of him.
Trent Alexander-Arnold - 2/10
He was virtually a non-entity for the duration of that game. Harrison completely got the better of him for the first Leeds goal, and the channel between him and Gomez was always a little susceptible.
Joe Gomez - 6/10
Gomez got sucked in to trying to win the ball back from Harrison too eagerly and got dribbled past, in the build-up to the first Leeds goal. But that aside, he was very capable on the cover and used his pace well to make a few key interceptions.
Virgil van Dijk - 6/10
The big Dutchman got his goal with a towering header after he was left unmarked in the box, but immediately returned the favour by gifting the ball for Bamford to slot past Alisson. He also had a goal disallowed in the second half, after a foul by Jones on Koch.
Andrew Robertson - 6/10
Robertson got himself his first assist of the season, with the corner for van Dijk's goal. He defended reasonably well against the likes of Pablo Hernandez and Helder Costa, even when he was put under tremendous pressure.
Jordan Henderson - 6/10
He won a few tackles in midfield, and his passing was generally really good. But it was clear that he wasn't yet fully fit, and left a lot of space behind him, which Leeds exploited really well.
Naby Keita - 5/10
He started the game brightly, but soon faded away. Some of his vertical passing was a real joy to watch, but he sometimes ended to try a little too hard in the face of the relentless Leeds press.
Gini Wijnaldum - 5/10
The Dutchman had a huge chance to put Liverpool in a 4-2 lead, but he struck a tame shot straight at Ilian Meslier in the Leeds goal. He was barely involved in midfield, and gave the ball away too cheaply once too often.
Mohamed Salah - 10/10
Salah became the first Liverpool player to score in the opening game of four consecutive league campaigns. He was outstanding in every aspect of his play, giving Stuart Dallas a fiendishly difficult ask of trying to defend against him. Salah was confident twice from the spot, and struck an absolute thunderbolt in between the two penalties on a night to remember.
Sadio Mane - 5/10
He was a constant outlet and threat down Liverpool's left, and really did put in a lot of work into finding the right spaces. But his decision making in the final third was atrocious. He denied Liverpool a potential early goal of the season contender, after a stunning passing sequence with Salah ended in him skying his shot.
Roberto Firmino - 4/10
He had no shots on target, and created no big chances in the game. Firmino also missed a sitter right at the end of the game, when the score still at 3-3, and could've regretted that, if not for Rodrigo's moment of madness right at the end.
Fabinho - 6/10
The Brazilian calmed things down in the middle of the park after replacing Keita. He won the penalty for the winning goal, after being fouled in the box by Rodrigo.
Curtis Jones - 5/10
He was confident in possession of the