A 2-0 win in Belfast last week in a match marred by missiles thrown at striker Leigh Griffiths had put Celtic in the driving seat heading back to Glasgow for the second leg.
And two goals from Scott Sinclair, along with one apiece from Rogic - who also scored in the first leg - and Stuart Armstrong, ensured a pain-free progression to the next round for Brendan Rodgers's men, who will take on Norwegian champions Rosenborg next.
The result stretches Celtic's unbeaten run to 35 matches, with the Scottish champions having last experienced defeat at home to Barcelona in the Champions League in November.
It took just four minutes for Celtic to open the scoring on the night, with Sinclair firing beyond Roy Carroll after an Olivier Ntcham effort deflected into his path.
Ntcham continued to pose a threat and Callum McGregor, Moussa Dembele and Rogic all went close in a first half totally dominated by the hosts.
Linfield will have felt fortunate to reach half-time just a goal down, but the floodgates opened early in the second period, with Rogic drilling in an effort from 25 yards two minutes after the restart.
Another fortuitous deflection played a part in Sinclair's second eight minutes later, but the Celtic winger was denied a hat-trick by Carroll when the former Manchester United goalkeeper tipped a curling shot onto his crossbar.
Armstrong wrapped up the victory in stoppage-time, burying his shot after good build-up play by Rogic.