The reigning Premier League champions were looking for a response to Saturday's last-gasp 2-1 loss at Crystal Palace but wilted on an intense evening in Istanbul.
Victor Osimhen's first-half penalty after Dominik Szoboszlai was adjudged to have caught Baris Alper Yilmaz with a trailing arm proved the difference at a rocking Rams Park.
Cody Gakpo had seen an effort cleared off the line moments before the decisive spot kick on an evening that saw Slot surprisingly drop Mohamed Salah and Alexander Isak to the bench.
The Liverpool boss turned to the pair in the 62nd minute in a bid to shake up his disjointed side shortly after losing goalkeeper Alisson Becker to an injury seemingly sustained when stopping Osimhen scoring a second.
Striker Hugo Ekitike also limped off as the Reds failed to bounce back from their Selhurst Park setback, with a late penalty for a foul on Ibrahima Konate overturned after a VAR review.
Elsewhere on Tuesday night, Kylian Mbappe scored a hat-trick and Harry Kane netted twice as European heavyweights Real Madrid and Bayern Munich enjoyed big wins over tournament newcomers.
Madrid shrugged off any travel weariness in a 5-0 win at Kairat Almaty in Kazakhstan and Bayern cruised to a 5-1 win at Pafos in Cyprus.
Atletico Madrid also romped to a 5-1 victory over Eintracht Frankfurt with coach Diego Simeone watching from the stands following his suspension for an altercation with a Liverpool fan in the team's previous Champions League game.
Jose Mourinho blew a kiss to Chelsea supporters on his return to Stamford Bridge with his Benfica team. An own goal from Richard Rios decided the result, 1-0 to the Londoners who had Joao Pedro sent off.
Tottenham needed a late own goal to escape Bodø/Glimt with a 2-2 draw from the Norwegian champion's first home game in the main stage of the competition.
Marseille beat visiting Ajax Amsterdam 4-0, Lautaro Martinez scored twice to lead Inter Milan to a 3-0 win at home over Slavia Prague and Atalanta came from behind to beat Club Brugge 2-1.
With PA.
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