The Scottish champions sit bottom of Group F with one point from three games after Christopher Nkunku and Andre Silva put them to the sword in Germany.
Nkunku looked to have broken the deadlock in the 17th minute when he chipped Hart, only to be ruled to have strayed offside by VAR.
Ten minutes later he did open the scoring, poking home after being played in by Silva after the Portugal forward's surge upfield on the counter.
Celtic levelled two minutes into the second half through Jota's first-time finish into the bottom-right corner but, after Dominik Szoboszlai saw a goal controversially chalked off for Silva being in an offside position, Leipzig restored its advantage in bizarre fashion.
Hart's pass out from the back was intercepted by Szoboszlai, who played in Silva for a simple finish.
Silva then doubled his tally by popping up at the back post to finish a fine team move and put the game to bed with 13 minutes remaining.