Leipzig took the lead 33 minutes in as winger Emil Forsberg raced down the left-hand side on the counter-attack before bursting into the box and smashing past Diego Benaglio at the near post.
Monaco levelled the game almost instantly from kick-off however, as Djibril Sidibe's cross was bundled over the line by Youri Tielemans for his first goal in Monaco colours.
Leipzig take the lead in their first Champions League match
The Bundesliga side made four changes to the side that beat Hamburger SV 2-0 on Saturday afternoon as Bruma, Jean-Kévin Augustin, Naby Keïta and Bernardo made way for Youssuf Poulsen, Lukas Klostermann, Stefan Ilsanker and Forsberg.
Leonardo Jardim made three changes to the side that fell to a 4-0 defeat against OGC Nice with Almamy Touré, Tielemans and Benaglio replacing Thomas Lemar, Rony Lopes and Daniel Subasic.
Ralph Hasenhüttl's men looked to impose themselves on the game in the opening 15 minutes and enjoyed the lions share of possession against a Monaco side who were set up to counter, but failed to create any clear chances until 24 minutes in.
A corner from the left hand-sde was whipped into the area and striker Poulsen rose highest to meet the ball, sending his header yards wide of Benaglio's goal.
Die Roten Bullen scored their first ever Champions League just after the half-hour mark, as Forsberg hit Monaco on the counter and ran free down the left-wing before blasting in at the near post, leaving keeper Benaglio with no chance.
Monaco's instant reply stuns the Red Bull Arena
Les Monégasques hit back almost instantly to stun the home crowd, as Sidibe's cross was nodded back across goal by Adama Diakhaby for summer signing Tielemans to prod past Péter Gulácsi and over the line.
Leipzig forward Timo Werner used his blistering pace to try and get in behind the Monaco back-line throughout the match, and nearly netted his first Champions League goal five minutes after the restart only for his shot from the edge of the box to narrowly miss Benaglio's goal.
Both sides struggled to impose themselves on the second-half as a flurry of fouls marred the game, with neither side offering much in the way of creativity in the final third.
Monaco's top-scorer last season, Radamel Falcao, cut an isolated figure for much of the game as service was limited with Leipzig the more dominant outfit.
The result sees Monaco sit in second place in Group G while Leipzig are third, with Beşiktaş topping the table thanks to their 3-1 win at Porto.