Celtic Park had noticeable gaps of empty seats, with some supporter groups having planned a boycott in protest against the board’s stewardship of the club in recent times.
O’Neill had been encouraged by the prospect of talks between fans and the club, while he spoke immediately before the match about a need to freshen his team up.
He made five changes to the starting XI and six substitutions during regulation time.
It took until Tounekti’s 70th-minute arrival for Celtic to really find their stride and on-loan Freiburg forward Adamu’s goal was one of the few occasions the hosts threatened in the six-yard box.
Tounekti’s winner, a low finish after a one-two with sub Luke McCowan, deflated Dundee and they scarcely looked likely to get the equaliser that would have resulted in penalties.
That said, Steven Pressley’s side did little wrong until Adamu’s goal and will count themselves unfortunate to be out of the cup after such a sizeable effort.
For O’Neill, a three-pronged focus remains with defending champions Celtic chasing leaders Hearts in pursuit of the Premiership title and a Europa League tie with Stuttgart on the horizon as well as continued involvement in the Scottish Cup, a trophy the veteran boss won three times between 2001 and 2005.
