The Police Service of Northern Ireland (PSNI) said “an extensive police investigation” had begun after the incident at the Greenvale Hotel in Cookstown, Co Tyrone, at around 9.30pm on Sunday.
Assistant Chief Constable Mark Hamilton, speaking from the scene, said: “Our preliminary investigations show there was a crush towards the front door of this hotel, and in that crush people seem to have fallen.
“There seemed to be a little bit of struggling going on to get people up off the ground and that might explain also why there was a report of some fighting.”
He made a direct appeal for people with footage of the incident to hand it to police, rather than post it on social media.
Mr Hamilton confirmed the three teenagers who died were a 17-year-old girl, a 17-year-old boy and a 16-year-old boy.
A 16-year-old girl is in a stable condition in hospital, while a further two teenagers were treated for injuries they suffered during the incident.
He added: “We are of an open mind ... it looks like a crowd of young people, a large crowd, pushing up against the front of the hotel to get in to this event.
“People then seem to have fallen and then got crushed. I would urge you to use that with some caution. Certainly it appears to be at the minute a crush, people have fallen and then been fatally injured.
“We are continuing to interview people who were there to establish the full facts and I would appeal to anyone who witnessed what happened to contact police on 101 ext 52014.
“We need parents to talk to their children this morning and encourage them to come and tell us what happened. Please do not post photographs or videos online. Please share them with the PSNI.”
He added: “We need people to come forward and tell us what happened last night. Three young people are dead, they deserve a full inquiry. Their parents need to know what happened to them.”
Police posted on Twitter shortly after the incident, urging parents to pick up their children from the venue in Drum Road.
Emma Heatherington, whose two sons, aged 17 and 18, had been at the event, said she “felt sick” when she saw the police appeal.
“My two boys were there so I rang them, shaking, and thankfully got them both,” she said. “They were on their way home at that stage.
“They say there was a huge crowd waiting to get into the venue and then a lot of commotion up near the front of the queue, then they said the emergency services arrived and they were all asked to leave.
“They came home very, very shaken with rumours that some people had died.”
The 42-year-old, of Donaghmore, Co Tyrone, added: “While it’s not clear exactly what happened, it’s the most tragic ending imaginable to what was supposed to be a celebration of St Patrick’s Day.”
A member of staff at the nearby Glenavon Hotel said he had heard the sirens as emergency services rushed to the scene, and said police had come to borrow their defibrillator.
“There were ambulances passing our entrance,” he said. “It’s scary, and you just hoped everyone was OK.
“It’s very sad to hear what happened.”
Colin Neill, chief executive of Hospitality Ulster, said: “We are devastated to hear of the confirmed PSNI report of at least two deaths this evening at the Greenvale Hotel in Cookstown. This is shocking and tragic news and our thoughts and prayers are with the families of the deceased.”
Democratic Unionist Party leader Arlene Foster tweeted that her “thoughts and prayers” were “with everyone impacted by this tonight”.
And Sinn Fein deputy leader Michelle O’Neill called it a “parent’s worst nightmare”.
“My thoughts and prayers are with the families,” she added.
Former Democratic Unionist MP for South Antrim William McCrea said: “We’ve got to see what lessons can be learned from this tragedy.
“I drove past this 10 minutes before it happened, not knowing that 10 minutes later an absolute tragedy would happen.
“The people that I’ve spoken to, they’re very shocked. It’s a close-knit community.”