Sinclair, who signed from Chelsea earlier this month, kept his cool to convert from close range after eight minutes as the Swans made it two wins from two on home soil in the Championship.
Darren Pratley was denied by the woodwork and later sent off on an eventful afternoon for the midfielder before Tyrone Mears was also shown a red card for two bookings.
Alan Tate headed wide inside the opening five minutes after meeting a David Cotterill free-kick and the hosts made the most of a promising start to surge ahead with eight minutes played.
Nathan Dyer exchanged passes with Pratley before producing a cross that Stephen Dobbie miskicked, but the ball fell kindly for Sinclair to net from close range.
Dobbie did connect properly with another Dyer centre shortly after, prodding wide from the edge of the six-yard box.
Swansea continued to dominate and were fortunate not to double their advantage when Pratley's 12-yard strike clipped the top of the crossbar, while Martin Paterson tested Dorus De Vries from 25 yards at the other end on a rare Burnley attack.
Paterson went close again from distance on the half-hour mark before Sinclair fired at Brian Jensen from just outside the penalty area.
Chris Iwelumo failed to hit the target with a close-range header from Graham Alexander's cross - and the former Wolves star and Dean Marney spurned further chances for the visitors just before the half-time interval.
The lively Cotterill blazed over from fully 25 yards after the restart and Pratley tried his luck from distance to no avail. Dyer and Sinclair combined again to allow the latter a sight at goal only for the former Chelsea trainee to fire wide.
Two red cards in the space of as many minutes then changed the complexion of the game. Pratley was the first to be dismissed for a challenge on Mears that earnt him his second booking of the afternoon.
Mears then received his marching orders after clattering Dyer and referee Steve Tanner wasted little time in brandishing a second yellow to the former Preston and Derby defender.
Steven Thompson replaced Paterson and almost hauled the sides level within four minutes of being introduced. Danny Fox crossed and Thompson saw his six-yard header saved by De Vries.
Jack Cork, on loan at Turf Moor from Chelsea, rifled over from 30 yards as Burnley searched for an equaliser and Thompson shot wide from a similar range during a frantic finish.
Andrea Orlandi went close at the opposite end and Ross Wallace crashed over a 25-yard free-kick, but the Swans hung on to take all three points.