Graeme Murty's side defended poorly as their rivals - the runaway Scottish Premiership leaders who have already won the League Cup - went 2-0 up through Rogic and Callum McGregor in the first half and then made the result emphatic thanks to penalties from Moussa Dembele and Olivier Ntcham, while Ross McCrorie was sent off.
Dembele set the tone for the afternoon when he crashed a volley on to the post after five minutes, connecting with Ntcham's floated cross, but Rangers held out until they allowed Rogic to control the ball and turn his man inside the penalty area before side-footing the ball into the bottom-right corner after 22 minutes.
The defending for Celtic's second goal was even worse, Russell Martin failing to clear Kieran Tierney's cross and instead presenting the ball to McGregor, who gladly dispatched it into the far corner of the net beyond the diving Wes Foderingham.
The win was all but wrapped up five minutes into the second half when Dembele burst down the left wing and cut inside before being pulled to the ground by McCrorie, who was shown a straight red card before the Frenchman chipped the penalty cheekily past Foderingham.
Craig Gordon made a superb one-handed save to deny Alfredo Morelos three minutes later and the Colombian was denied again when he dwelled on the ball in a penalty box scramble and saw his shot smothered.
With 12 minutes left to play, Rangers substitute Jason Holt felled Patrick Roberts in the box and Ntcham drilled the ensuing spot-kick into the bottom-left corner to put the gloss on a result that never looked in doubt from the moment Celtic took the lead.