The Ligue 1 champions were bottom of the group prior to kickoff with just two points from their four games so far, and got off to a nightmare start against their opponents.
On the half-hour mark, Monaco's hopes of remaining in the competition were effectively killed off as Radamel Falcao's foul on Willi Orban in the box was punished with a penalty, which Werner duly rattled home from the spot.
No sooner had the Monaco fans gained a sliver of hope, however, than the visitors restored their three-goal advantage as Naby Keita turned Kamil Glik superbly and fired into the bottom corner past Danijel Subasic on the stroke of half time.
Leonardo Jardim's charges set about trying to salvage some pride from the game and came close just before the hour, when Joao Moutinho's free kick was flicked on by Keita Balde and almost turned in from close range by Andrea Raggi.
However, Leipzig were able to see the game out without any problems, condemning last season's Champions League semi-finalists to fourth place and not only an early exit from this year's competition, but also from the Europa League.