Die Werkself created the better chances, only to be undone by Joao Moutinho's predatory finish in the 61st minute.
Calhanoglu to the fore
Facing Ligue 1 opposition for the first time since being knocked out of last season's competition by Paris-Saint Germain in the Round of 16, Leverkusen - unbeaten under Roger Schmidt - started brightly enough. Hakan Calhanoglu combined neatly with Heung-Min Son to release Gonzalo Castro, but the German midfielder was unable to direct his effort on target.
Calhanoglu was everywhere for the visitors, rifling wide from distance before planting an inch-perfect free kick on the head of Ömer Toprak, who glanced over. The Turkish international playmaker dragged a decent chance wide before Son followed suit towards the end of a first half in which Leverkusen asked all the questions of a hitherto timid Monaco ensemble.
Schmidt's men continued to force the issue after the restart, though clear-cut opportunities were proving decidedly hard to come by. The hosts, in contrast, had offered virtually nothing as an attacking force, but got their noses in front on the hour-mark when former Leverkusen striker Dimitar Berbatov's cushioned header found Moutinho, who stroked home with the aid of a deflection off Emir Spahic.
Calhanoglu was the first man to respond, but Bellarabi arrived a fraction too late after Monaco goalkeeper Danijel Subasic could only parry the lively Leverkusen No10's fierce drive back out into the danger zone. Unable to summon the equaliser in the dying stages, die Werkself will have to wait until their 1 October showdown with SL Benfica, who lost 2-0 to FC Zenit St. Petersburg in the group's other contest, to get their European campaign off the ground.
Monaco: Subasic, Fabinho, Kurzawa, Carvalho, Raggi, Moutinho, Kondogbia (Wallace 90+4'), Toulalan, Berbatov, Ocampos (Bernardo Silva 57'), Ferreira-Carrasco (Dirar 86')
Leverkusen: Leno - Jedvaj (Donati 65'), Toprak (Reinartz 71'), Spahic, Boenisch - Bender (Drmic 76'), Castro - Calhanoglu, Son - Bellarabi, Kießling
Goals: 1-0 Moutinho (61')