Earlier, Manchester United eased the pressure on Erik ten Hag with a comfortable 3-0 win at Southampton.
Forest had not won at Anfield in the top flight for 55 years but the home faithful were stunned when Callum Hudson-Odoi fired in the winner 18 minutes from time.
Liverpool had won all three of their opening games under Slot without conceding, including a 3-0 win away at Manchester United two weeks ago.
But the Reds were sloppy and sluggish after the international break and made to pay.
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