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Hearts stunned Rangers for the second time this season to knock them out of the Scottish Cup in a fiery quarter-final at Tynecastle.

Rangers, who were without Alfredo Morelos after he was dropped due to a disciplinary issue, suffered defeat away to the Premiership's bottom side after also doing so last month.

Oliver Bozanic drove Hearts ahead from a defensive mix-up, and they could have increased their lead in a strong showing that also saw Steven Naismith go close late on.

Hearts, who had a goal disallowed in the first half for an extraordinary Loic Damour handball, will find out their opponents in Sunday's semi-final draw, and they could be paired with Edinburgh rivals Hibernian after they beat Inverness Caledonian Thistle on Friday.

However, the result came at a cost with defender John Souttar suffering a season-ending Achilles injury after just 16 minutes after falling to the ground off the ball, with Scott Arfield and Ryan Jack forced off early for the visitors.

Whether it was the arrival of storm Jorge or the fact that there were many empty seats in the home end inside Tynecastle, the atmosphere was very flat in the opening period.

But with Rangers looking lacklustre up front in the absence of Morelos, the benched Jermain Defoe and the cup-tied Florian Kamberi, Hearts twice came close to opening the scoring.

Steven Naismith forced Allan McGregor into a fingertip save with a header from eight yards out, and Damour had a volley blocked from close range.

Referee Steven McLean was having a difficult night and Rangers midfielder Jack may well have been lucky to only see a yellow after a high challenge on Michael Smith.

But just as the first half was petering out the game burst into action.

As Naismith raced towards the Rangers goal, defender George Edmundson clearly used his left arm to help usher the ball away from danger. However, the incident was missed by the referee.

The controversy continued from the corner when Damour used both hands to flick the ball beyond the visitors defence and into the bottom corner of the goal. MacLean initially awarded the goal, triggering hails of protests from the Rangers players. But McLean changed his mind and gave the visitors a free-kick after consulting his officials. However, he elected not to give the Frenchman a second yellow card, and on the half-time whistle Steven Gerrard and Naismith argued their points at the mouth of the tunnel.

Early in the second half Damour had a volley deflected wide by the knee of Edmundson as the visitors defence continued to look shaky.

The home side grew in confidence, with the breakthrough coming giftwrapped.

James Tavernier struggled to clear, Lewis Moore chased what looked to be a lost cause, cut the ball back for Bozanic after McGregor had committed himself, and the ball was hammered home from close range.

Rangers pushed forward, leaving gaps at the back. And, after a mistake by Edmundson on the edge of the box, Naismith was given a clear view of goal but the ball sliced off the edge of his boot and wide of the target.

The visitors did have chances with Edmundson twice heading over from around six yards out, but the Hearts defence stood firm to earn their place in the semi-final.

Hearts manager Daniel Stendel: "I'll give the same answer as Thursday. This was the game which was important for today, and Tuesday is the game that's important for the league. In the end we need more victories in the league, but the start is that we have this behaviour, especially in the defence and we score goals that win games."

Rangers manager Steven Gerrard: "On Wednesday night I was the proudest man in Europe because to a man my players were outstanding. Everything we have worked on for nearly two years I could see it on the sidelines and I was proud as punch. It was an incredible performance.

"But today for 90 minutes I didn't recognise anything."