Uefa delayed kick-off by more than 30 minutes, citing “security reasons”.
The match at the Stade de France in Paris did not start until 20:36 BST, with Real Madrid Beating Liverpool 0-1.
Police outside the ground fired tear gas as a small number of supporters tried to climb over security barriers.
Liverpool defender Andy Robertson said the organisation of the match was a “shambles”.
While Merseyside Police’s matchday officers tweeted that it was “the worst European match I’ve ever worked or experienced”.
They added: “I thought the behaviour of the fans at the turnstiles was exemplary in shocking circumstances. You were not late 100%.”
Pictures had emerged in the build-up of huge crowds of fans queuing to get in.
Many claimed they had been at the stadium hours before kick-off but were stopped from getting into the ground.
“We are hugely disappointed at the stadium entry issues and breakdown of the security perimeter that Liverpool fans faced this evening at Stade de France,” the Anfield club said in a statement.
“This is the greatest match in European football and supporters should not have to experience the scenes we have witnessed tonight.
“We have officially requested a formal investigation into the causes of these unacceptable issues.”
European football’s governing body Uefa said it was “sympathetic to those affected by these events and will further review these matters urgently together with the French police and authorities, and with the French Football Federation”.
“In the lead-up to the game, the turnstiles at the Liverpool end became blocked by thousands of fans who had purchased fake tickets which did not work in the turnstiles,” it explained.
“This created a build-up of fans trying to get in. As a result, the kick-off was delayed by 35 minutes to allow as many fans as possible with genuine tickets to gain access.
“As numbers outside the stadium continued to build up after kick-off, the police dispersed them with tear gas and forced them away from the stadium.”
French police had urged fans not to force entry with tweets 20 minutes before the game was scheduled to begin.
As kick-off approached, large areas of empty seats could still be seen inside the ground.
Outside, some fans were seen climbing perimeter fences or attempting to run past security. Riot police were also in attendance.
Liverpool players emerged to warm up again at 20:06, with messages on the big screens inside the stadium stating the delay was because of the late arrival of fans.
French police said fans with fake tickets had tried to force access to the stadium but “the rapid intervention of the police allowed the return to calm”.
They added they were able to disperse fans “without difficulty”.
They also said there were no major incidents at fan zones in the city, although 500 people were moved on from a bar near the Liverpool fanzone at Cours de Vincennes.
‘A completely unreasonable situation’ – the fans’ view
BBC journalist Nick Parrott, who was outside the ground in a personal capacity, told BBC Radio 5 Live: “I have just been pepper sprayed for the first time in my life. I have been to the Champions League final before and I have never been pepper sprayed before.
“I was outside Gate Y which has been opening and closing repeatedly over the last hour and a half but nobody has been going through. The security staff on the other side advanced and sprayed pepper spray.”
The opening ceremony started at just after 20:20 BST before the game finally got under way 16 minutes later.
Some 10 minutes after kick-off, Football Supporters Europe said “thousands are still trapped outside the stadium, remaining calm in the face of a completely unreasonable situation”.
The body added: “Fans at the Champions League final bear no responsibility for tonight’s fiasco.”
BBC Sport’s John Watson, at the ground, said he had “passed hundreds of Liverpool fans queuing at entry gates an hour before kick-off and not being permitted entry”.
“I’m at a loss to understand the reasoning for the delay being the late arrival of fans,” he added.
Liverpool West Derby MP Ian Byrne tweeted to say he had “endured one of the worst experiences in my life”, blaming Uefa for “horrendous security and organisation putting lives at risk”.
“I pray no fans have been injured because of the disgraceful lack of organisation and expertise,” he added.
Dominic King, from the Daily Mail, told BBC Radio 5 Live he had been speaking to supporters queueing to get into the stadium.
“It came from nowhere,” he said. “There has been no violence or antagonism. They are just dismayed by how the authorities have treated them.”
“We got told about 45 minutes before, that our end was filled while there were thousands outside,” Liverpool fan Tyler, who spent £1,000 on his ticket, told BBC Radio 5 Live.
The final was originally to be played in St Petersburg, but the city was stripped of the game following Russia’s invasion of Ukraine.