Celtic beat Rangers on penalties to edge into the Scottish Cup semi-finals as a combustible Old Firm derby concluded in ugly scenes with fans from both sides swarming the pitch at Ibrox.

James Tavernier and Djeidi Gassama missed from the spot after an absorbing - if poor quality - contest, allowing Tomas Cvancara the chance to seal a place in the last four after a game in which Rangers dominated.

When his penalty beat Jack Butland, Celtic fans spilled on to the field, followed by Rangers supporters, as police and stewards tried to restore order while projectiles flew through the air.

Police Scotland condemned the "shameful" scenes which they say included "extreme hostility and violence over a sustained period".

They said some fans were armed with "items intended to cause harm" and that officers and members of the public have been injured and "a number of arrests" have been made.

Both clubs told BBC Sport they will not comment officially yet, but the Scottish FA said "an investigation will be carried out immediately".

It was an ugly end to an engrossing quarter-final.

Both sides had a goal ruled out in the 90 minutes, with Celtic's Daizen Maeda having a header disallowed for an earlier offside before Emmanauel Fernandez saw his effort chalked off for handball in extra time.

Rangers dominated for long spells but failed to capitalise on that as Celtic held firm to earn an incredible victory, perfectly executing their four spot-kicks to claim a place in the semi-final draw, later on Sunday.

The headlines, however, will focus on the aftermath. It was horrible to witness and will surely bring consequences.

Before the horrible ending, it was tense. A quality contest, no. Dramatic, absolutely. Nerve shredding for all invested in it.

The absence of Callum McGregor and Kieran Tierney through injury handed Celtic manager Martin O'Neill a real selection dilemma.

Liam Scales moved to left-back, Benjamin Arthur made just his second start, and Luke McCowan largely occupied McGregor's position from the start.

All that surely offered Rangers a huge pre-match boost. They asserted themselves as the first half wore on but struggled to threaten Viljami Sinisalo in Celtic's goal.

The visitors then got a foothold and thought they had scored when Maeda's flying header found the target, but Scales was deemed offside on review as his flick on created the chance.

A let off for Rangers but no direct shots on goal from either side by the interval told the main story.

After the break, the hosts threw everything at Celtic, who looked to be creaking. Arthur was caught and bailed out by Sinisalo. Mikey Moore was twice denied, superbly, by Auston Trusty.

Pressure built. Fernandez flicked on a corner, Youssef Chermiti got a touch but Sebastian Tounekti somehow scrambled clear.

Scramble being the operative word for Celtic, they were having to constantly. The home side kept coming with O'Neill's side backed up against a wall in their own half.

Benjamin Nygren denied Nicolas Raskin on the goal line but Celtic forced extra time showing dogged determination to not relent.

Still Rangers came. Chermiti hit one from a great position but was blocked. From the corner, Fernandez rose, the ball was forced home but was it handball? It was, as the drama continued.

The chance rate for Rangers upped but the scoreline somehow remained blank.

Celtic seemed content to hold out for spot-kicks. They got them and somehow escaped with victory.

Whoever the victor, they would automatically become favourites to lift the Scottish Cup. Although, in this season of shocks nothing seems a given.

Given the current Premiership context, this route to silverware was not one to be treated lightly. It might be the only trophy available to these clubs.

Celtic were badly stretched. Rangers nearly full strength.

What an opportunity this was for Danny Rohl's side. They had Celtic on the ropes but, as has been the case too often of late, they couldn't capitalise when in a great position.

Rohl has said he needs to lift silverware this season. That's the demand at Rangers and for too long that has largely escaped them. That is a long shot now.

This was undoubtedly a patched up Celtic team. Crucially, a clutch of new signings were left out, perhaps suggesting recruitment hasn't been up to scratch again.

In attack, Celtic were almost anonymous. Incredibly, they had one shot - and that off target - in 120 minutes. You don't win games like that. Yet they did.

Defensively, they were quite something, though.

They had won in Stuttgart and Aberdeen recently. Drawn at Ibrox last weekend when seemingly on the floor. Somehow, here, they got the job done against all odds.

The scenes as they celebrated will be dissected and dealt with but Celtic are through and remain in with a shout of silverware.

Rangers head coach Danny Rohl: "We're disappointed. I saw a team from our side to play 120 minutes how we want to. We should score.

"When you have 24 to one shots, when you create so many great opportunities, then I must say, I'm proud of my group, how we played football. I know football's about results. I see a lot of positive things."

Celtic manager Martin O'Neill: "Well, obviously delighted that we're still in the competition.

"The players showed phenomenal courage and great, great determination and a rear guard action, which is really commendable. Everyone to a man did more than their jobs. They were really terrific."

Both sides resume their Scottish Premiership title tilts.

Celtic host fourth-placed Motherwell on Saturday (15:00 GMT) and Rangers visit St Mirren on Sunday (12:00).