After a fairly uneventful and goalless first 45 minutes, the Parkhead side took the lead in the 53rd minute when Gorgie midfielder Oliver Bozanic tripped Ryan Christie in the penalty area allowing fellow substitute Scott Sinclair to slam in the penalty.
In a unique cup semi-final at the home of Scottish rugby, James Forrest capitalised on a mistake by Gorgie keeper Zdenek Zlamal to fire in an easy second.
Christie scored a wonderful third to set up a final appearance at Hampden Park on December 2 against Aberdeen or Rangers, who play at the national stadium later on Sunday.
Celtic boss Brendan Rodgers has won six out of six domestic trophies since taking over at Parkhead in 2016 and this was a significant step towards a treble treble.
Much of the build-up to the game had been about the key players missing from both sides.
Rodgers made six changes to his side which started the 2-0 defeat to RB Leipzig in the Europa League on Thursday night.
Defender Dedryck Boyata was suspended but Forrest returned from a ban while central defenders Filip Benkovic and Kristoffer Ajer, midfielder Tom Rogic, defender Mikael Lustig and goalkeeper Scott Bain, in goal for the previous two ties, were reinstated.
Hearts boss Craig Levein made two changes with midfielders Olly Lee and Peter Haring in for Harry Cochrane and Callumn Morrison, who started on the bench amid a fantastic atmosphere.
Referee Willie Collum booked striker Steven MacLean in the first minute for a foul on Lustig and then skipper Steven Naismith went off injured seven minutes later to be replaced by Danny Amankwaa.
Nothing much of note happened at either end before Hoops midfielder Eboue Kouassi came off injured in the 26th minute, his place taken by Sinclair.
MacLean had the ball in the Hoops net in the 40th minute following a Lee corner which had been headed goalwards by Haring but was correctly given offside.
The Parkhead side raced up the other end and Sinclair’s drive from the edge of the box was parried behind by Zlamal for a Callum McGregor corner which came to nothing.
Christie took over from Olivier Ntcham for the start of the second half and within seven minutes he had won a penalty when he was tripped by Bozanic, referee Willie Collum pointing to the spot and Sinclair sending Zlamal the wrong way with the spot-kick.
Celtic stepped up a gear and moments later, French striker Odsonne Edouard headed a Kieran Tierney cross over the bar.
Zlamal was at fault for Celtic’s second goal, making a hash of saving a tame
long-range effort from Christie and when he scooped the ball out to Forrest he easily returned it into the net from close range.
Christie made it 3-0 in the 72nd minute when he guided a terrific shot high past Zlamal after picking up the loose ball 25 yards from goal.
The Parkhead side were relentless.
Edouard’s curling shot from 12 yards beat Zlamal but defender Jimmy Dunne cleared off the line before the Hearts keeper made saves from Forrest and Tierney - and Sinclair was twice denied by Gorgie defenders - but by then it was academic, Celtic were on their way to Hampden again.