Match ends, Heart of Midlothian 4, Motherwell 2.
Late goals from Olly Lee and Steven Naismith sent Hearts into their first semi-final since 2014.
Craig Levein's league leaders will face holders Celtic after extending their unbeaten run to 12 games.
They trailed to Curtis Main's penalty, but dominated thereafter, with Steven MacLean prodding in an equaliser and Peter Haring putting them ahead.
Ryan Bowman thumped Motherwell level, but Hearts kept up the pressure and scored twice in the dying minutes.
Hearts have not won the competition since 1962, but now face a Celtic side who they beat 1-0 at Tynecastle earlier in the season.
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Fortress Tynecastle
Tynecastle has been something of a fortress for Hearts in recent times with only Celtic being able to claim victory over Hearts on their home turf in the last 16 months.
And the way Levein's side started, they certainly looked determined that sterling home record would remain intact.
Within the opening minutes, both Naismith and Haring had headers cleared off the line by Curtis Main.
But Motherwell always looked dangerous, and Bowman knew exactly what he was doing when he peeled off the back of the Hearts defence to nod the ball back across goal. It hit the arm of Jimmy Dunne and Main made no mistake from the penalty spot.
However, when you have players at your disposal like Naismith and Arnaud Djoum then you always have a chance, and both played their part in the equaliser with MacLean getting the final touch.
Haring had glanced a header just inches wide moments before he took full advantage of another ball into the box and knocked it over the line from close range.
The game though was far from over and Bowman thundered a stunning volley high into the roof of the net with Zdenek Zlamal given no chance in the home goal.
But just as it looked as if extra-time was on the cards, the home side regained the lead with Lee's neat finish finding the net from the centre of the box.
Then Naismith showed great composure for Hearts fourth, leaving Tom Aldred in his wake and calmly sliding the ball low into the net. A fine goal to finish off an amazing game of football.
Hearts a match for any team - analysis
The strength of Hearts playing in front of their own supporters cannot be underestimated. Whenever Motherwell looked as if they were about to take control, the Tynecastle roar became ever so noticeable.
But Motherwell more than played their part in a compelling game. And, if manager Stephen Robinson is able to keep Bowman and Main fit, it will not be too long before they climb the table.
But this was Hearts' night and in Naismith they surely have captured one of the contenders for player of the season even at this early stage. He was constantly talking to the youngsters in the team and, in particular, Callumn Morrison who certainly looks as if he has the potential to go far.
The win means Hearts equal their third-best ever start to a season, and on this form they will be more than a match for any team in the country.
Line-ups
Hearts
Formation 4-3-3
- 1Zlamal
- 2Smith
- 4Souttar
- 3Dunne
- 11MitchellBooked at 28mins
- 8LeeSubstituted forBozanicat 90+2'minutes
- 10Sutchuin-Djoum
- 5Haring
- 38MorrisonSubstituted forDikamonaat 86'minutes
- 18MacLeanSubstituted forAmankwaaat 90+2'minutes
- 14Naismith
Substitutes
- 7Bozanic
- 12Amankwaa
- 13Doyle
- 17Garuccio
- 21McDonald
- 23Mulraney
- 28Dikamona
Motherwell
Formation 3-5-2
- 1Carson
- 5Aldred
- 22Donnelly
- 4McHughBooked at 67mins
- 7Cadden
- 8CampbellSubstituted forJohnsonat 75'minutes
- 17BigirimanaSubstituted forFrearat 66'minutes
- 14Grimshaw
- 2Tait
- 9Main
- 12Bowman
Substitutes
- 3Taylor-Sinclair
- 11Frear
- 20Gillespie
- 21Mbulu
- 24Johnson
- 28Turnbull
- 30Maguire
- Referee:
- William Collum
- Attendance:
- 14,377
Match Stats
- Possession
- Home63%
- Away37%
- Shots
- Home22
- Away10
- Shots on Target
- Home8
- Away4
- Corners
- Home12
- Away5
- Fouls
- Home13
- Away15
Live Text
Post update
Full Time
Second Half ends, Heart of Midlothian 4, Motherwell 2.
Post update
Corner, Heart of Midlothian. Conceded by Richard Tait.
Substitution
Substitution, Heart of Midlothian. Danny Amankwaa replaces Steven MacLean.
Substitution
Substitution, Heart of Midlothian. Oliver Bozanic replaces Olly Lee.
Goal!
Goal! Heart of Midlothian 4, Motherwell 2. Steven Naismith (Heart of Midlothian) right footed shot from the centre of the box to the bottom left corner. Assisted by Michael Smith.
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Corner, Motherwell. Conceded by Demetri Mitchell.
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Corner, Motherwell. Conceded by John Souttar.
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Attempt saved. Olly Lee (Heart of Midlothian) right footed shot from outside the box is saved in the centre of the goal.
Goal!
Goal! Heart of Midlothian 3, Motherwell 2. Olly Lee (Heart of Midlothian) right footed shot from the centre of the box to the bottom right corner. Assisted by Michael Smith.
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Foul by Jimmy Dunne (Heart of Midlothian).
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Curtis Main (Motherwell) wins a free kick in the defensive half.
Substitution
Substitution, Heart of Midlothian. Clévid Dikamona replaces Callumn Morrison.
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Liam Donnelly (Motherwell) wins a free kick in the attacking half.
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Foul by Djoum (Heart of Midlothian).
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Attempt missed. Djoum (Heart of Midlothian) right footed shot from the centre of the box is high and wide to the right.
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Corner, Heart of Midlothian. Conceded by Liam Donnelly.
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Steven Naismith (Heart of Midlothian) wins a free kick in the defensive half.
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Foul by Carl McHugh (Motherwell).
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Attempt missed. Steven Naismith (Heart of Midlothian) header from the centre of the box is just a bit too high.