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Robbie Neilson wants Hearts to "make some history" after becoming the first team through to this season's Scottish Cup semi-finals.
Beni Baningame, Peter Haring, Aaron McEneff and Ellis Simms scored in Saturday's 4-2 win against St Mirren.
Eamonn Brophy's striker and Connor Ronan's stunning 30-yarder had brought the visitors level at 2-2.
"We want to win football matches, we want to win cups," Neilson told BBC Scotland.
"We want to try and make some history. We've still got two more games to go."
Hearts won the Scottish Cup for the eighth time in 2012 and last reached the final in 2020.
The draw for the last four takes place after Monday's tie between Dundee United and Celtic, with Motherwell hosting Hibernian and Rangers visiting Dundee in Sunday's quarter-finals.
Hearts were without regular starters John Souttar and Cammy Devlin and listed youngsters Mackenzie Kirk, Luke Rathie and Macaulay Tait on the bench.
And there was a worrying moment for manager Neilson in the opening minutes when Craig Halkett collided with Brophy and had to have a head gash bandaged up. Brophy also received treatment.
The home side soon settled and took the lead after Barrie McKay found Simms in the six-yard box. The striker could not force the ball past Jak Alnwick but Baningime arrived to apply a firm finish - his first Hearts goal.
St Mirren responded by loading the Hearts box at a throw-in and Scott Tanser's dangerous-looking shot ricocheted off a couple of bodies and prompted faint calls for handball.
Jordan Jones then crossed low for Greg Kiltie but he could not keep his effort down, to the relief of the home defence. Manager Stephen Robinson later said there had been a foul on Kiltie after he took the shot.
But the visitors were undone by a sweeping move as Liam Boyce found Ben Woodburn on the right and his square pass was knocked high into the net by Haring.
Robinson's side responded again and this time were able to find the net in style. Richard Tait sent a lovely ball in from the right and Brophy's outstretched foot carried it past Craig Gordon.
Gordon had to look lively at the visitors' next attack as Brophy's head connected with a Jones free-kick.
A hugely entertaining first half had another big moment when Halkett rattled the St Mirren crossbar after Boyce's overhead kick had been blocked.
Alnwick was called into action twice in quick succession after the interval, blocking Nathaniel Atkinson's shot then shifting direction to palm over a deflected Woodburn effort.
Gordon was equally astute to get strong hands behind Jones' shot then Kiltie's header.
But the Scotland goalkeeper could only look on in horror as Ronan collected a cleared corner on the right, around 30 yards from goal, and lifted his shot into the opposite corner. A truly incredible strike.
It was game on again and Andy Halliday played a one-two with Boyce before drawing a brave save from Alnwick.
McEneff was introduced in place of Woodburn and arrived at just the right moment to score Hearts' third. Atkinson did well to get away from Jones and cut back, with the ball deflected into McEneff's path for him to improvise a shot with his thigh.
Haring headed over and Stephen Kingsley's free-kick was touched wide by Alnwick as Hearts finished strongly, culminating in Simms converting a McEneff delivery at the front post to finally put the result beyond doubt.
Man of the match - Peter Haring
What they said
Hearts manager Robbie Neilson: "It was a great game, a proper cup tie. I was right behind that second goal for St Mirren - I don't think I've seen a better one in my time here. Really pleased to get through.
"Over the course, the chances created, clear cut chances, we probably deserved to win. We want to win football matches, we want to win cups. We want to try and make some history. We've still got two more games to go."
St Mirren manager Stephen Robinson: "We got back to 2-2 with an absolute wonder goal. You need to hold it for five, 10 minutes, calm the crowd down. We didn't do the basics right in the defensive side of things for the third goal.
"We should've had a penalty. It looks like a stonewall penalty on Greg Kiltie. Ultimately, if you concede four goals, you're not going to win the game. We ran out of a little bit of steam in the last 15 minutes."
What's next
Both clubs return to Scottish Premiership action next Saturday with home matches, Hearts facing Livingston and St Mirren hosting Dundee United (both 15:00 GMT).
Player of the match
HalkettCraig Halkett
with an average of 7.66
Heart of Midlothian
Squad number19Player nameHalkett
Squad number8Player nameMcEneff
Squad number6Player nameBaningime
Squad number1Player nameGordon
Squad number5Player nameHaring
Squad number18Player nameMcKay
Squad number10Player nameBoyce
Squad number3Player nameKingsley
Squad number12Player nameAtkinson
Squad number20Player nameSimms
Squad number16Player nameHalliday
Squad number9Player nameWoodburn
Squad number21Player nameSibbick
St Mirren
Squad number43Player nameRonan
Squad number9Player nameBrophy
Squad number13Player nameGogic
Squad number6Player namePower
Squad number7Player nameJones
Squad number2Player nameTait
Squad number21Player nameGreive
Squad number12Player nameHenderson
Squad number18Player nameDunne
Squad number1Player nameAlnwick
Squad number11Player nameKiltie
Squad number22Player nameFraser
Squad number3Player nameTanser
Squad number4Player nameShaughnessy
Line-ups
Hearts
Formation 4-2-3-1
- 1Gordon
- 12Atkinson
- 19Halkett
- 3Kingsley
- 16Halliday
- 5Haring
- 6BaningimeBooked at 40mins
- 9WoodburnSubstituted forMcEneffat 67'minutes
- 10Boyce
- 18McKaySubstituted forSibbickat 90+3'minutes
- 20Simms
Substitutes
- 8McEneff
- 13Stewart
- 15Moore
- 21Sibbick
- 44Kirk
- 47Rathie
- 49Tait
St Mirren
Formation 4-5-1
- 1Alnwick
- 2Tait
- 4Shaughnessy
- 18Dunne
- 3TanserSubstituted forFraserat 53'minutes
- 11KiltieSubstituted forHendersonat 75'minutes
- 43Ronan
- 6Power
- 13GogicSubstituted forGreiveat 75'minutes
- 7Jones
- 9Brophy
Substitutes
- 5McCarthy
- 8Flynn
- 10Main
- 12Henderson
- 16Erhahon
- 21Greive
- 22Fraser
- 26Lyness
- 39Smith
Referee:Bobby Madden
Attendance:13,899
Live Text
Match ends, Heart of Midlothian 4, St. Mirren 2.
Second Half ends, Heart of Midlothian 4, St. Mirren 2.
Substitution, Heart of Midlothian. Toby Sibbick replaces Barrie McKay.
Attempt missed. Beni Baningime (Heart of Midlothian) right footed shot from outside the box is too high.
Goal! Heart of Midlothian 4, St. Mirren 2. Ellis Simms (Heart of Midlothian) left footed shot from very close range to the top right corner. Assisted by Aaron McEneff.
Corner, Heart of Midlothian. Conceded by Jak Alnwick.
Attempt saved. Stephen Kingsley (Heart of Midlothian) left footed shot from outside the box is saved in the top right corner.
Foul by Alex Greive (St. Mirren).
Nathaniel Atkinson (Heart of Midlothian) wins a free kick in the defensive half.
Attempt blocked. Ellis Simms (Heart of Midlothian) right footed shot from the left side of the box is blocked.
Attempt missed. Peter Haring (Heart of Midlothian) header from the centre of the box is close, but misses to the right.
Corner, Heart of Midlothian. Conceded by Richard Tait.
Attempt missed. Craig Halkett (Heart of Midlothian) header from the centre of the box misses to the left.
Substitution, St. Mirren. Alex Greive replaces Alexandros Gogic.
Substitution, St. Mirren. Jay Henderson replaces Greg Kiltie.
Foul by Jordan Jones (St. Mirren).
Liam Boyce (Heart of Midlothian) wins a free kick in the defensive half.
Attempt missed. Peter Haring (Heart of Midlothian) right footed shot from the centre of the box is too high.
Attempt blocked. Ellis Simms (Heart of Midlothian) right footed shot from the centre of the box is blocked.
Goal! Heart of Midlothian 3, St. Mirren 2. Aaron McEneff (Heart of Midlothian) with an attempt from the centre of the box to the top right corner.