The Hoops forward squeezed in the opener in the third minute before skipper Scott Brown thundered in a 30-yard drive six minutes later.
Returning winger James Forrest added a third in the 52nd minute, with Sinclair scoring twice more in a one-sided affair toseal thecup holders' place in Monday night's quarter-final draw.
Celtic had beaten Tommy Wright's side 2-0 both home and away in the Ladbrokes Premiershipbefore their midweek win over Hibernian and it was their fifth victory of the season over the Perth side, who are on the slide after five successive defeats.
Brendan Rodgers' team have now won seven matches out of seven since the turn of the year, scoring 21 goals and conceding none as they turn their attentions to Thursday night's first leg of a Europa League tie against Valencia.
The visitors' chances of pulling off a cup shock ended almost before they had started.
When striker Oliver Burke burst all too easily through the Perth defence it looked like he made an error in squaring the ball instead of shooting as Sinclair was at an acute angle.
However, the Hoops winger beat Zander Clark at his near post, with the McDiarmid Park keeper looking culpable.
What remaining belief Saints had - and it could not have been much - evaporated when Brown picked up a clearance and arrowed a drive high past the flailing Clark for his first counter in two years.
The visitors struggled to cope with Celtic's movement and fluidity.
However, just after the half-hour mark, Parkhead keeper Scott Bain had to throw himself to his right to push away a Chris Kane drive and the home side defended consecutive corners.
Wright swapped midfielder Sean Goss with Tony Watt for the start of the second half while Kristoffer Ajer replaced Celtic centre-back Jozo Simunovic.
The visitors tried to press Celtic high but that left gaps and Burke was sent racing clear by a wonder Ryan Christie pass, but he looked off-balance as his weak attempt at a shot went straight to Clark.
Moments later in another swift Celtic counter, however, left-back Jonny Hayes, in for injured Emilio Izaguirre, sped through the middle before laying the ball off to Forrest, who slotted in from 12 yards.
Two minutes later, after Forrest had set up Burke, Sinclair tapped in from close range.
A rout similar to the 6-0 win over Saints in October was threatened and it should have been five when Christie headed over a Forrest cross.
Defender Niall Keown, on loan from Partick Thistle, replaced winger Matty Kennedy to make hisSt Johnstone debut as the visitors sought to limit any further damage.
Christie made way for French striker Odsonne Edouard and, in the final minute, after the latter's shot fell to him inside the Saints box, Sinclair slammed his third in from 10 yards, taking his tally to an impressive 16 so far this season.
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