The Merseyside club dominated proceedings in the first half with Philippe Coutinho impressing, but the game was turned on its head in the second half when he was replaced.
Substitute Shinji Okazaki opened the scoring in the 66th minute before Slimani fired in from outside the box to seal victory for the Foxes.
Jurgen Klopp's side, which made the semi-finals past year and final in 2016, has now failed to win any of its last four matches.
Both sides fielded slightly weakened sides but there was still enough on the pitch to ensure it was a competitive game. That is okay, but then you need to stay concentrated.
"You are in a situation with your old club - and I don't say this about Alex specifically - and you accept it and it is hard to make the next step". They scored too easy.
It upset Liverpool's momentum but Klopp insisted the Brazil global was always down to come off. I'm sick of goals like this to be honest, you can not play football and do that.
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"I thought it made complete sense for Alex to change club and to come here", Klopp said. "He needs match time but it was not the plan to let him play again 60 or 70 or 80 minutes", said Klopp.
'From today's perspective yes, we have the resources, but we will see how it is again in January'.
"The first goal, we don't push up together.
Leicester made it two-nil 13 minutes later when Okazaki found Slimani and the Algeria global charged towards goal before smashing the ball into the top corner.