Match ends, Heart of Midlothian 0, Celtic 3.
Celtic booked their place in the Scottish League Cup final thanks to a debatable penalty, a goalkeeping error and a 20-yard stunner that all arrived in 19 second-half minutes.
First Scott Sinclair scored from the spot when Ryan Christie went down under Oliver Bozanic's challenge.
Hearts' Zdenek Zlamal fumbled a tame Christie shot and James Forrest put away the rebound to double the lead.
Christie then stroked home a superb third goal for the cup holders.
Celtic now will meet Aberdeen - who beat Rangers 1-0 in Sunday's second semi-final - in the 2 December showpiece at Hampden as Brendan Rodgers targets a seventh domestic trophy in a row.
Celtic get job done when they need to
The prelude to this tie was all about the players Celtic were missing - captain Scott Brown, centre-backs Dedryck Boyata and Jozo Simunovic, and goalscorer Leigh Griffiths.
But it was Hearts who lost their leader and fulcrum after just seven minutes, Steven Naismith pulling up off the ball with a leg problem and limping off.
The Premiership leaders were lucky not to lose Steven Maclean, too, with the striker surreptitiously grabbing a very delicate part of Eboue Kouassi's anatomy at a corner having already been booked.
The first half was a bit of a grind, but it sputtered into life near the end, Maclean having an effort disallowed after bundling in while several yards offside, before Zlamal tipped a vicious Sinclair shot past.
Eight minutes into the new half, the holders had their breakthrough, but in contentious fashion.
Christie jinked past Bozanic, then hit the deck. It was a glancing touch, but enough for referee Willie Collum to point to the spot.
Sinclair stuttered, sent Zlamal the wrong way, and stroked the ball low into the net for his third goal of the season.
Celtic had the lead and they began to motor. Their passing was crisper. Tom Rogic, an imperious presence at the heart of it all, pirouetted and loped and swaggered forward with growing menace.
They duly got their second goal, although it came from a horrible Zlamal mistake. The Czech allowed Christie's tame 18-yard shot to slip beneath him. He pawed it back out from the goalline - it might, in fact, have crept over - but Forrest was waiting regardless to gleefully slam the ball beyond the goalkeeper.
There was nothing Zlamal could do about Celtic's third, though. Christie ran on to Odsonne Edouard's flick, and from just outside the box, arrowed a sublime effort high into the top net. Game over.
Edouard had chances to make it four. Zlamal redeemed himself a little with a miraculous stop to thwart Forrest, then parried clear a Kieran Tierney strike. Sinclair also had two shots cleared off the line.
A day that began with such hope and vigour for Craig Levein's Premiership leaders ended with Celtic rampant.
Again, domestically, they got the job done when it mattered. Again, in a huge cup tie played against a thunderous backdrop, facing a team at the top of their game, they delivered.
'Plaudits must go to Rodgers' - analysis
BBC Scotland's Brian McLauchlin at Murrayfield
The early action failed to live up to the magnificent atmosphere generated by the 60,000 plus fans inside Murrayfield but there is no doubt Hearts losing Naismith was pivotal.
But the plaudits must go to Celtic manager Rodgers. His decision to replace Eboue Kouassi with Scott Sinclair was a major turning point and the introduction of Ryan Christie was huge, too. The Hearts rearguard was completely unable to cope with his pace and directness.
In the second half, Celtic looked like the team who have lifted six domestic trophies in succession. And it will be a warning to their opponents that they were able to put three past the league leaders.
Hearts will have to pick themselves up quickly as neighbours Hibernian come calling in midweek. But with a number of players carrying knocks, this is when they will find out the real depth of their squad.
- 10Sutchuin-DjoumBooked at 87mins
- 8LeeSubstituted forClareat 76'minutes
- 7BozanicSubstituted forMorrisonat 70'minutes
- 14NaismithSubstituted forAmankwaaat 8'minutes
- 18MacLeanBooked at 2mins
- 35AjerSubstituted forHendryat 80'minutes
- 88KouassiSubstituted forSinclairat 26'minutes
- 21NtchamSubstituted forChristieat 45'minutes
- William Collum
- Shots on Target
Second Half ends, Heart of Midlothian 0, Celtic 3.
Corner, Celtic. Conceded by Michael Smith.
Attempt blocked. Scott Sinclair (Celtic) right footed shot from the centre of the box is blocked.
Attempt blocked. Scott Sinclair (Celtic) left footed shot from the centre of the box is blocked.
Attempt blocked. Odsonne Edouard (Celtic) left footed shot from the centre of the box is blocked.
Corner, Celtic. Conceded by Michael Smith.
Attempt blocked. Kieran Tierney (Celtic) left footed shot from outside the box is blocked.
Corner, Celtic. Conceded by Clévid Dikamona.
Attempt blocked. Callum McGregor (Celtic) left footed shot from outside the box is blocked.
Attempt saved. Sean Clare (Heart of Midlothian) right footed shot from outside the box is saved in the centre of the goal.
Attempt saved. Odsonne Edouard (Celtic) right footed shot from outside the box is saved in the centre of the goal.
Djoum (Heart of Midlothian) is shown the yellow card for a bad foul.
Tomas Rogic (Celtic) wins a free kick in the defensive half.
Foul by Djoum (Heart of Midlothian).
Attempt blocked. Odsonne Edouard (Celtic) right footed shot from the centre of the box is blocked.
Foul by Sean Clare (Heart of Midlothian).
Callum McGregor (Celtic) wins a free kick in the defensive half.
Foul by Filip Benkovic (Celtic).
Danny Amankwaa (Heart of Midlothian) wins a free kick on the right wing.