It got worse for the visitors as lightning struck twice before the break; Joel Matip combined with Salah and gave Leeds keeper Illan Meslier no chance with the finish, before Sadio Mane was fouled inside the box by Luke Ayling for Salah to claim his brace.
Three goals to the good and without any signs of danger in the second half, it was a matter of cruise control for Klopp's side as they were patient with the ball, and it was Mane who eventually wrapped up the game with a brace in the final 10 minutes of the game to make it five.
Virgil van Dijk's bullet header rounded off a 6-0 win in added time.
Liverpool were clinical with the chances and could have scored five in the first half. It was a perfect performance ahead of Sunday's Carabao Cup final against Chelsea.
An all-round solid performance for the Reds means there aren't any glaring negatives, though Salah could have been more clinical in front of goal from open play.
Manager rating out of 10
Jurgen Klopp, 10 -- A positive team selection put Liverpool into a commanding position early on, while Klopp's game plan created all sorts of problems for Marcelo Bielsa, securing the biggest home win of his Liverpool career.
Player ratings (1-10; 10 = best, players introduced after 70 minutes get no rating)
DF Andy Robertson, 7 -- The Scotland captain stretched the play down the left flank and it was his cross that was handled by Dallas to win a penalty for Liverpool.
DF Virgil van Dijk, 8 -- The 30-year-old reacted quickly to the danger and was often the first to intervene when Leeds attempted to play forward. A well-taken header put the game to bed to complete the victory.
DF Joel Matip, 8 -- Matip looked assured at the back and capped off his performance with a finish taken in the manner of a seasoned striker.
DF Trent Alexander-Arnold, 7 -- A precise ball put Salah clean through on goal just before the break, with Alexander-Arnold unlucky not to pick up an assist from the Egypt international's tame finish.
MF Fabinho, 7 -- The former Monaco midfielder wasn't called upon as much as usual on the night, but he will be wondering how he didn't score as he completely scuffed his effort wide of goal close to half-time.
MF Thiago, 8 -- Rarely got a pass wrong in a game and looked in his element as he calmly moved the ball around the pitch. Thiago was also solid defensively, breaking up the play a number of times in midfield areas.
MF Curtis Jones, 8 -- Jones played with invention and a will to get the ball forward while never panicking under pressure. The 21-year-old could have scored at the hour mark as he found his way into the box but very unselfishly picked out Salah, who fired over.
FW Luis Diaz, 8 -- The 25-year-old was a problem all game for the Leeds United defence, regularly challenging defenders and using his pace on counter-attacks to get into dangerous areas. While he should have done better with his chance that went straight at the goalkeeper, he looked liked he enjoyed himself at Anfield.
FW Sadio Mane, 9 -- A well-timed run won the second penalty of the night for Liverpool's third before Mane scored two goals of his own in the final 10 minutes.
FW Mohamed Salah, 8 -- Dispatched both penalties with ease, but his assist for Matip's goal was even better, perfectly guiding the ball into the defender's path for his first goal of the season. Should have done better with his chances to complete the hat trick.
MF Jordan Henderson, 6 -- Introduced for Thiago to play the final 25 minutes and picked out Mane inside the box for Liverpool's fourth goal.
MF James Milner, N/R -- Kept things simple in his run out against the club that he made his Premier League debut for all the way back in 2002.
FW Divock Origi, N/R -- On for the final five minutes in place of Diaz and involved in the fifth goal as his shot was diverted into the path of Mane.