Sinclair scored twice in three minutes in the first half to put Brendan Rodgers' side in charge. Stuart Armstrong then made it 3-0 in the 55th minute before Patrick Roberts curled in a fine fourth.
Sinclair completed his hat-trick from the penalty spot late on as Celtic won the second part of a potential treble, with a Scottish Cup semi-final against Rangers to come on April 23.
The win also sees the Hoops break a 100-year club record of going 37 domestic games unbeaten to start a season, and they are on track to become the first unbeaten Scottish champions since Rangers in 1899.
Hearts started the game brightly, with Esmael Goncalves threatening as he shot at Craig Gordon and then had another shot saved from a tight angle.
But, after Callum McGregor missed a brilliant chance from six yards out for Celtic, Sinclair opened the scoring when he played a one-two with Roberts and fired home emphatically into the roof of the net in the 24th minute.
It was 2-0 shortly afterwards as Roberts slipped in Sinclair and he finished the one-on-one chance.
The quickfire double seemed to knock the stuffing out of Hearts, who saw their challenge firmly fade away after half-time.
Armstrong scored Celtic's third when he finished from the edge of the box before Roberts curled in a well-struck shot into the top corner.
Dedryck Boyata headed over as Celtic controlled the closing stages, with Sinclair completing the rout when he finished from the penalty spot after being fouled by Krystian Nowak.