Celtic have apologised to fans caught up in a crush outside the club's stadium.
One person was taken to hospital and four more were injured before Sunday's match against Rangers after stadium access points were changed.
The club said it regretted that so many fans were faced with the situation.
Police Scotland have promised a "thorough and comprehensive" debrief of the policing and stewarding operation.
Celtic said the match was the first occasion which featured new segregation and access arrangements for the Old Firm fixture.
Some fans said the new plan resulted in some supporters being forced to climb over a high fence to escape the overcrowding, with one falling from a wall.
The statement, posted on the club's website, said: "The safety of our fans will always be of paramount importance and this is a matter which we are treating with the utmost seriousness. The Club, along with Police Scotland, will be investigating this matter fully to identify the causes and take the appropriate steps to ensure this does not happen again at Celtic Park.
"We are fully aware of the serious difficulties which arose yesterday for supporters, and we sincerely regret and apologise that so many fans were faced with this situation."
The statement added: "Yesterday's match was the first occasion which featured new segregation and access arrangements.
"Our total focus now is to work with the police in looking at these arrangements and other circumstances surrounding this particular fixture and to take the correct action to ensure this situation is never repeated.
"Our priority will always be to ensure our supporters enjoy a safe and positive experience at Celtic Park."
Police Scotland responded to criticisms of its operation by promising to conduct a full review.
Ch Supt Brian McInulty said on Monday: "Following the events that took place before yesterday's Celtic v Rangers SPFL match, tomorrow I will be meeting with representatives of Celtic FC and the other emergency services to undertake a thorough and comprehensive de-brief of the policing and stewarding operation.
"Later this week we will also be happy to meet with representatives of Celtic FC supporters groups to discuss any concerns they may have."
'Tried and tested plans'
He continued: "The safety of all those attending matches remains the absolute priority for Police Scotland.
"The situation yesterday was a fluid one and decisions were taken quickly to prevent any further persons from gaining access to Janefield Street and to alleviate the congestion in this area.
"The response to the overcrowding was in line with tried and tested plans involving stewards, police officers on foot and mounted officers."
Ch Supt McInulty also addressed speculation regarding a gate having been closed on Janefield Street and contributing to the congestion.
He said: "At this time we believe that the gate was open at all times in the lead up to the match, however as this has been raised as a concern I will ensure that this forms part of the review".
The incident happened about 20 minutes before the 12:00 kick-off as Celtic supporters tried to make their way into the stadium.
Police had earlier cordoned off a section of London Road to allow Rangers' 800 fans access, forcing more home supporters than usual to use the Janefield Street entry point.
'Worry and panic'
Hundreds of people were then caught in a two-way crush in the corridor under the stadium's North Stand.
Celtic fan Jamie Kerr was in the middle of the crush.
He told the BBC: "We arrived around 11:30. As we walked into the tunnel, it did feel pretty crammed. But we didn't think anything of it because it's a busy football match.
"But things took a really bad turn when more and more people were coming into the tunnel and nobody was moving forward so the crowds were coming in from both directions.
"It was actually very, very frightening. The good nature of the tunnel very quickly turned into one of worry, one of panic. There were kids crying."
He added: "Even the looks on the faces of some of the fathers with their children. People could not move. And the problem was more people were coming in. I am surprised there were not very serious injuries."
Labour MSP for Glasgow, James Kelly, has called on Police Scotland to explain the reasons for the change in normal policing arrangements.
He said: "This is a very frightening situation.
"There clearly needs to be serious questions answered by Police Scotland as to why we ended up in this situation.
"This ground regularly hosts 60,000 capacity crowds. We have never seen an incident like this outside the ground. Why did it happen yesterday? Very worrying indeed.
"We need to understand why that gate was closed off when it is not normally closed off."
Celtic supporter groups have also called for a thorough investigation.
Jeanette Findlay, chairwoman of the Celtic Trust, told BBC Radio Scotland's John Beattie: "The stories we have heard are extremely distressing.
"We have heard very detailed accounts of numerous failures of both police and stewards and the completely distressing affect on lots of fans - elderly people crying, people having panic attacks - really serious distress and a failure to act.
"Policing is key to this because when the situation develops in real time the police have the responsibility to take control of that as they would in any crowd situation.
"What we have is a culture inside the police service of Scotland which is to treat the fans as though they are potential criminals or sources of disorder. And so the culture is thinking about and planning for games in that way.
"It's not about thinking, here are a group of citizens who are in a crowd situation and we need to keep them safe."
The Celtic Trust said it would be consulting with Celtic FC on Monday, and called for fan groups to be involved in any debrief with the club and police.
The match ended in a 1-0 victory for Celtic.