The 33-year-old will begin his European assignment when Leverkusen open their campaign at home against Slovenian side NK Celje.
The Bundesliga club will then encounter several teams from outside Europe's traditional top-five leagues during the league phase, although Ligue 1 pair Marseille and Lyon are also among their opponents.
Flekken's selection comes despite a difficult start to his Bundesliga campaign, having shipped five goals in Leverkusen's opening two away fixtures.
"As for the goalkeeper, I have decided that Mark (Flekken) will play on Wednesday," coach Carles Martinez said.
"Mistakes are part of football. That's something I firmly believe in. If someone makes a mistake, that's no reason to say, 'come on, you're out!'"
"I think he's a great professional. He's doing incredible things. He's the first one to be there. He's the first to push for us to get better as a squad. So I am very satisfied in many ways. My philosophy is to stand by my players, especially after mistakes."
The decision represents a change from the goalkeeping arrangement used during Xabi Alonso's time in charge. Matej Kovar was the preferred option for cup matches under the former coach, while Lukas Hradecky occupied the starting berth in Bundesliga fixtures.
Martinez's decision therefore establishes a clearer split between Leverkusen's European and domestic cup goalkeeping responsibilities.
