Portugal international Moutinho, one of Monaco's big signings last year, fired home from Dimitar Berbatov's header in the 61st minute.
Monaco, who sold Colombia international Rodriguez to Real Madrid and loaned compatriot Falcao to Manchester United, had been second best Bundesliga leaders Leverkusen for the first hour.
The principality side's poor domestic form was reflected in an indifferent opening with Karim Bellarabi making a hash of a great chance to fire the visitors ahead from Hakan Calhanoglu's defence-splitting pass.
Leverkusen were dominant in midfield but toothless up front with Heung-min Son sending a low shot wide when left unmarked in the box.
On the stroke of halftime, Gonzalo Castro also managed to fire a volley from close range over the bar as the visitors continued to waste chances.
Monaco were struggling to keep Leverkusen at bay and Layvin Kurzawa and Yannick Ferreira Carrasco were both booked before the break, which Monaco were grateful to reach at 0-0.
It was Leonardo Jardim's side, however, who went ahead shortly after the hour.
Bulgarian striker Berbatov headed a long ball into the path of midfielder Mountinho, whose shot was deflected by Emir Spahic's heel past Bernd Leno.
Leverkusen had their first shot on target 10 minutes from time when Castro's fierce shot was parried away by Danijel Subasic.
"We knew that it would be difficult against a good team who have been brilliant in their league," Monaco midfielder Geoffrey Kondogbia told BeIN Sport.
"But we also knew we had the weapons to beat them. We were solid. They had a lot of chances but we also had our opportunities."
(Writing by Julien Pretot; Editing by Justin Palmer)
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