Video circulating on social media showed security scuffling with fans as they ran across the concourses and attempted to breach the gates of the Hard Rock Stadium in Miami Gardens, Florida.
Both teams were taken off the field as the chaos brewed outside.
Police managed to close the gates and initiate a lockdown, leading to scores of people stuck outside trying to enter before the match kicked off.
South American football's governing body CONMEBOL pushed the match's start time back three times from 2:00 am CET to 2:30 am CET to 2:45 am CET and finally to 3:15 am CET, when the teams were finally able to line up for the national anthems.
"We inform that people who do not have tickets will not be able to enter the stadium," CONMEBOL said on X, formerly Twitter.
"Only those who have purchased tickets will be able to enter the stadium once the entrances are reopened."
ESPN reported that for at least one gate, fans with tickets were being allowed to enter one at a time.
"In anticipation of tonight's Copa America final, thousands of fans without tickets attempted to forcibly enter the stadium, putting other fans, security and law enforcement officers at extreme risk," a statement from a Hard Rock Stadium spokesperson said.
"Security has shut the gates in order to control the entry process at a much slower rate and ensure everyone is kept safe."
Miami-Dade police said there were several incidents before the gates were opened at the stadium.
"These incidents have been a result of the unruly behaviour of fans trying to access the stadium," they said on social media. "We are asking everyone to be patient, and abide by the rules set by our officers and Hard Rock Stadium personnel."
FOX quoted a police officer as saying the situation had not improved outside the stadium.
CONMEBOL said they had tightened security following the semi-final between Uruguay and Colombia, when Uruguayan players clashed with Colombian fans after the match ended.
They also confirmed that Shakira's half-time show in the final would last 20 minutes. Colombia coach Nestor Lorenzo had said he disagreed with the extended half-time break beyond the usual 15 minutes because it could affect players' fitness.
