Substitute Lucas Ocampos scored with 18 minutes to play as Leverkusen - who completely dominated the Group C contest on Wednesday - were left to rue their profligacy in front of goal.
Argentinian Ocampos was introduced just two minutes before his winner and his strike lifted Monaco to second in the pool.
The French side now have eight points and are just one behind Leverkusen, who could have qualified with a win or a draw.
With Zenit in third on seven points, Monaco's win means three sides from the pool can still qualify for the round of 16 heading into matchday six - when Monaco host Zenit and Leverkusen go to bottom-placed Benfica.
Tricky South Korean winger Son Heung-min had a fierce effort blocked by Ricardo Carvalho as Leverkusen began to dominate, and only the woodwork prevented the BayArena crowd from celebrating in the 28th minute.
Lars Bender was the unlucky player, smashing a volley against the crossbar after Monaco had failed to clear their lines.
Leverkusen head coach Roger Schmidt introduced Josip Drmic just before the hour mark in a bid to find a goal, but the Switzerland international hurriedly shot wide when presented with a good chance.
Calhanoglu lost the ball on the wing and Berbatov showed great skill before sliding a pass into Nabil Dirar, who swung in an outstanding cross from the right.
Dirar's cross beat Giulio Donati and fell perfectly for Ocampos, who slid in to volley past goalkeeper Bernd Leno.
Drmic then wasted a one-on-one opportunity, shooting at Subasic, and his miss summed up Leverkusen's night - one of wasted chances and disappointment.
