Liverpool made it three wins from three in the Champions League with a tight 1-0 victory away at RB Leipzig thanks to Darwin Nunez’s first-half tap-in.
Against a club that will soon become a concern of former Reds boss Jurgen Klopp, Slot's men set two new club records - the first Liverpool side to win their first six away games of a season and the first to triumph in 11 of their first 12 matches.
Nunez, brought into the starting XI in place of the injured Diogo Jota, delivered the winner in the 27th minute, poking home Mohamed Salah's header, and a seventh clean sheet of the campaign made sure of victory.
Top of the Premier League, and with nine points from three Champions League games, Liverpool's results have been outstanding, but after Slot warned he wanted to see his side do a better job of dominating games there were plenty of sticky patches against a Leipzig team who had two goals disallowed.
Leipzig, who sit level on points with Bundesliga leaders Bayern Munich, started as the better side and were on top at the end too, but could not avoid a third straight Champions League defeat.
Bar a dramatic turnaround Klopp could find that the jewel in Red Bull's footballing crown are facing an early exit from the competition by the time he starts as their head of global soccer in January.
Yet they could have led just minutes into the match. After a lapse from Kostas Tsimikas, starting in place of Andy Robertson, Xavi Simons lifted a shot over.
Trent Alexander-Arnold then gave the ball away, but an unmarked Benjamin Sesko headed straight at Caoimhin Kelleher.
Kelleher stopped a drilled low shot from Amadou Haidara but Liverpool's stand-in goalkeeper almost got it all wrong moments later, involved in a mix-up with Ibrahima Konate and then grateful to see Sesko bend his shot wide as he scrambled back.
Lois Openda was clearly offside when he fired home in the 26th minute, but the disallowed goal seemed to finally wake Liverpool up.
A minute later, the visitors were ahead. Tsimikas chipped in a cross from the left and Salah headed goalwards before Nunez stole in to prod it over the line.
Momentum shifted immediately. Nunez was denied a second by Leipzig goalkeeper Peter Gulacsi, the Hungarian who sat on Liverpool's bench 51 times early in his career without ever making an appearance.
Nunez wanted a penalty in the 38th minute after being tripped by Leipzig skipper Willi Orban. Replays showed contact but Swiss referee Sandro Scharer was not convinced and VAR somehow did not intervene.
Cody Gakpo shot into the side-netting at the end of the half and had an even better chance four minutes after the break, somehow denied by Gulacsi after Lutsharel Geertruida - Slot's former captain at Feyenoord - and Arthur Vermeeren got in each other's way dealing with a Nunez cross.
Just before the hour Salah cut in from the right hand side but bent his shot over. A few minutes later, the Egyptian was unhappy to be substituted - no doubt with Sunday's trip to Arsenal in mind - as Luis Diaz came on.
Alexis Mac Allister then saw a bending shot bounce off the crossbar as Liverpool continued to spurn chances to kill the game off, and it almost proved costly.
First Kelleher did well to save from Sesko after Konate lost the ball. Moments later, the Irishman tipped over a dipping shot from Simons.
Leipzig had the ball in the net again seven minutes from time, but Openda was offside as he bundled in the rebound after Kelleher got a hand to Benjamin Henrichs' cross, and Liverpool hung on.
Additional reporting from PA
