The Ligue 1 club headed into Tuesday's Group C clash on the back of a dismal start to their Ligue 1 season, with Leonardo Jardim's side having taken just four points from their first five matches to sit second-bottom of the table.
Some Monaco fans have demanded their money back from season ticket purchases after Radamel Falcao and James Rodriguez were sold in the transfer window.
There was sparse crowd for their Champions League opener at Stade Louis II, but those who did turn out were treated to a victory in the Principality club's first game in Europe's premier club competition since 2005.
It was Portugal international Moutinho who made the difference, settling an otherwise uninspiring game against Roger Schmidt's Bundesliga leaders with a 61st minute goal to secure three precious points.
Chances proved few and far between in a quiet start, although it was Leverkusen dictating the play in the early stages.
Despite enjoying a dominance of possession, the visitors were unable to threaten Danijel Subasic in the Monaco goal.
Leverkusen midfielder Gonzalo Castro fired well wide in the 13th minute for the visitors, wasting the first real opening of the game, then Hakan Calhanoglu shot into the side-netting two minutes later.
That was as good as it got for either side until the 36th minute, when Leverkusen winger Karim Bellarabi could only scuff his strike after being played through by a perfectly placed ball over the top from Calhanoglu.
Schmidt's side continued to show a startling profligacy in front of goal and spurned another glorious opportunity to break the deadlock when Son Heung-min failed to capitalise on sloppy Monaco defending, hitting a dismal effort way off target.
Castro then went close in first-half stoppage time, with his volley deflected over the crossbar by Layvin Kurzawa after Monaco had failed to deal with a Leverkusen free-kick.
The second half began in similarly dour fashion to the first, yet the game was brought to life when Moutinho finally opened the scoring.
Ex-Porto midfielder Moutinho latched on to a deft knock down from striker Dimitar Berbatov - playing against one of his former clubs - to finish with an effort that deflected off defender Emir Spahic over goalkeeper Bernd Leno and into the net.
That goal gave the hosts the confidence to press for a second and, although they were unable to double their advantage, the win should serve as a considerable morale boost for Jardim's men in their bid to kick-start their Ligue 1 campaign.
