Ange Postecoglu's side were aiming to avoid a repeat of last week's 3-1 defeat at the Red Bull Arena, but goals from Timo Werner and Emil Forsberg after the break decided a lively contest, with the Bhoys struggling to make inroads despite enjoying plenty of touches in the final third.
An all-action opening period saw Daizen Maeda spurn a good chance with less than a minute gone, but Leipzig began to control proceedings after that missed header.
However, Celtic were then cursing their luck as Matt O'Riley and Greg Taylor rattled the woodwork within a few seconds of one another, as the former's long-range strike cannoned off the post before Taylor's looping follow-up hit the crossbar before Leipzig managed to get the ball to safety.
Kyogo Furuhashi then headed just over the bar in acres of space as the Bhoys began to ramp up the pressure, but their momentum was halted with Liel Abada was forced off through injury at the end of the first half - James Forrest replaced him for his 450th Celtic appearance.
Joe Hart was forced to stick out a leg to keep out a Christopher Nkunku drive before the half-time whistle blew, as Celtic were left to rue their missed chances during the break.
Both sides continued their gung-ho styles of play in the embryonic period of the second half, albeit without the net rippling once, and Moritz Jenz was called into action to block a goal bound effort from Werner.
Leipzig then screamed for a penalty when Nkunku charged forward and went down under a challenge from Hart, but the Celtic goalkeeper got enough of the ball to avoid giving away a spot kick.
However, Die Roten Bullen would need no penalty to take the lead with 15 minutes remaining, as Andre Silva darted into space in the left-hand side of the box before picking out Werner, whose header into the far corner left Hart with no chance.
Celtic desperately tried to keep their Champions League last-16 hopes alive, but Postecoglou's side could not fashion any real opportunities to level the scores, and Leipzig were out of sight in the 83rd minute.
Werner was heavily involved again, bringing down a long ball with a delightful touch before picking out substitute Forsberg inside the area, and the Sweden international took a touch before picking out the top corner - leaving Hart flat-footed.
There was no consolation to be had for Celtic, who remain winless with just one point from four games in Group F, while Leipzig are up to second after Real Madrid levelled late on against Shakhtar Donetsk.