The visitors were dominant from the off and they took a deserved lead in the 17th minute following a corner, Sinclair's tame header wrong-footing Linfield goalkeeper Roy Carroll and creeping into the net.
Linfield held their shape well during the opening stages of the match, but defending set pieces proved problematic, and they fell further behind from another corner in the 22nd minute.
Leigh Griffiths played a low ball into the box and Australian international Rogic struck it with a crisp half-volley, beating Carroll for the second time in five minutes to double the Scottish side's lead.
Brendan Rodgers's men continued to pile on the pressure and were unlucky not to take a third goal into the break as Carroll produced a pair of impressive saves to twice deny Sinclair.
Linfield's best chance to pull a goal back against the run of play in the first period came in the 25th minute when Mark Stafford blazed Chris Casement's cross well over from close range, but such moments were rare.
Celtic had the ball in the net again early in the second half when Griffiths smashed home from close range, but the Scotland international was denied by a raised offside flag.
David Healy's side showed more attacking intent in the second period and they created a few openings of their own amid constant Celtic pressure.
Carroll was the standout player for the hosts, pulling off a string of additional saves in the second period to keep the deficit down, including an impressive reaction stop to deny Stuart Armstrong on the volley at close quarters.
The two sides will meet again at Parkhead next Wednesday, with the winner of the tie due to play either Dundalk or Rosenborg in the next qualifying round.