The Scottish champions took the lead just after the half-hour mark when Australian international Tom Rogic beat two defenders before squaring a pass into the box, and it was deflected into his own net by Evgeniy Postnikov.
Sinclair netted his second and Celtic's third on the hour, with Leigh Griffiths picking the former Aston Villa man out with a perfectly-weighted pass before he shot beyond the goalkeeper for a second time.
Griffiths was the provider again when the Hoops racked up their fourth, holding the ball up for James Forrest to strike from the edge of the box, the midfielder finding the net from a tight angle.
The Scotland international added his own name to the scoresheet in the closing stages, cutting inside the box and letting fly, his shot sailing straight into the far corner.
The result sees Celtic claim a joint-record winning margin and a record home win for the qualifying stages of the Champions League.