A supporter managed to enter the field of play and confronted Gers captain James Tavernier on the stroke of half-time before being arrested.
After the match, Dempster insisted “nothing was off the table” in terms of action the club could take after two unsavoury incidents in successive matches.
In a lengthy statement posted on the club’s website, Dempster said: “For the record, I did not “suggest” the closure of the East Stand, as has been claimed.
“Asked a question, I gave an answer. Nothing is, or can be, off the table but it is clearly not my intention to immediately or negatively impact on the vast majority of well-behaved supporters who sit in that area.”
Dempster confirmed that other measures would be deployed “before any closure is considered”.
Reinforcing her request that Hibs fans help identify troublemakers in the support, Dempster continued: “Hibernian – your club – has been the focus of much debate and national attention in recent days for entirely the wrong reasons.
“Supporters will share my dismay that a minority are damaging the proud reputation of this wonderful club - founded as it was on principles of charity, tolerance, and inclusion.
“I spoke of my anger at the conclusion of our recent match against Rangers at Easter Road.
“When we should have been talking about an exciting game and a performance of real character by our team, we have instead spent our time discussing the loutish behaviour of individuals who seem determined to ruin football for all of us.
“It has to stop and we need your help to make Easter Road Stadium the vibrant, noisy, colourful, exhilarating but safe, welcoming and tolerant place it normally is.
“Of course, the problems are not confined to this club. To suggest otherwise is a nonsense but we first must look after our own home and our reputation by facing into the issues we have experienced.
“Please, help us crack down on unacceptable behaviour. If you witness something that causes you concern during the match, let a steward know or write in to us, providing us with as much detail as you can.
“We will act.”
The Hibs chief also confirmed that the spectator who confronted Tavernier “will never attend a game at Easter Road again”.
She continued: “He is not a season ticket holder, though he was on our database and was able to purchase a match ticket. We have immediately applied the limit of the sanctions available to us.
“This means that his account has been closed, he has been added to a list of those banned from the ground and every effort will be made to ensure he never attends any Hibernian match or event again.
“We can say no more. He is currently being dealt with through the courts and we do not want to prejudice any sanction he may receive.”
The club has also suspended two other season-ticket holders who were both arrested “for different reasons” according to Dempster, who wrote: “The information we have is that both were arrested for different reasons - one for allegedly using a pyrotechnic outside the stadium and the other for allegedly attempting to get trackside after the equaliser.
“Again, they were detained by Police Scotland on the night, so it is unwise for us to say any more at this stage.
“Those who misbehave taint us all, so I want to make a point of thanking the supporters who attended Easter Road to support their team and who enjoyed a thrilling and competitive match.”
Dempster also defended the club’s CCTV network, pointing out that the existing system had been installed within the past three years and was “specifically designed for the stadium by a qualified security and safety officer”.
The former Motherwell chief executive continued: “I would argue it is a match for any system other than those at Celtic Park and Ibrox. It has nine cameras inside the stadium and 14 outside, all of which are monitored in the matchday control room by our own security staff and police. They direct the cameras, which allows for detailed live-time monitoring and recording.
“The purpose of upgrading is to add a further layer, which will allow us to record, in high definition, key areas of the stadium throughout the match.
“This approach is being tested now and a new camera was in play for the game on Friday night. Once installed the additional cameras will put our system on par with the best available.”
Dempster also confirmed the club’s Safety and Security Officer, Alan Marshall, had been asked to contact his colleagues in the Football Safety Officers Association of Scotland with a view to meeting to discuss what “effective and workable solutions” could be reached.
Hibs also have plans to contact many of the club’s supporters, via email and matchday letters, in a bid to “reinforce the need for supporters to behave in a way that does the club proud”.
Dempster continued: “In this debate one key principle must be agreed by everyone. We, as individuals, are responsible for how we choose to behave. It is our responsibility.
“When we make our behaviour the responsibility of others, we inadvertently condone those who choose to conduct themselves badly.
“Finally, I want to thank the many people who work incredibly hard to deliver a safe and enjoyable match day at Easter Road – your efforts have been tarnished by a few unpleasant incidents.
“We will not tolerate loutish antisocial behaviour.
“If you conduct yourself in this way I warn you that we will act to the limit of our influence to remove you from our game.”