The on-loan Celtic forward was filmed launching the pyrotechnic with his left foot after it had been thrown on to the field following a St Johnstone goal.
The incident sparked a police investigation, while Griffiths is almost certain to face retrospective punishment from the Scottish FA. It is not believed anyone was injured.
In a club statement in which Dundee addressed the “regrettable incident”, the 31-year-old insisted he was only trying to knock the flare off the pitch and had no intention of it travelling into the away crowd as they chanted abuse at the Scottish international.
Griffiths was quoted as saying: “It was regrettable that the pyrotechnic ended up back in the stand as my intention was just to remove it from the pitch.
“Having just lost a goal I was eager to get the match restarted as quickly as possible and I would like to apologise for any distress caused by this action.”
Dundee have said they will make no further comment at this moment in time.
It’s the second time this year Griffiths has been investigated by Police Scotland after he was cleared of any wrongdoing this summer following the publication of alleged text messages sent to teenage girls.
James McPake’s men were beaten by the reigning holders as goals from Shaun Rooney and Ali Crawford in the last 20 minutes helped the Perth side to reach the semi-finals.