The Hoops' pre-match preparations were overshadowed by a training-ground incident that left defender Emilio Izaguirre requiring hospital treatment after a violent clash with team-mate Nadir Ciftci.
But there was no sign of disharmony in the Parkhead ranks as they shrugged off a disjointed display to grind out a 2-1 win.
Leigh Griffiths had been a frustrated spectator for much of the match but showed lethal composure to slot his team ahead on 71 minutes before creating the second.
Rogic - who was introduced to the game on 59 minutes - got on the end of a cross from Griffiths to find the next in style with a clever back-heel on 82 minutes.
CLICK HERE to watch Rogic goal
Arnaud Djoum headed a late consolation effort for the Jambos but it is the cup holders who will now take their place in Monday's draw for the last four.
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