Argentina have been crowned Copa America champions after a dramatic final which was delayed by more than an hour and was highlighted by images of Lionel Messi crying on the bench.

Lautaro Martinez's extra-time winner earned the reigning world champions a 1-0 over Colombia in a match marred with drama and controversy. 

Argentina was able to win the Copa America for legendary captain Messi, who limped off with an ankle injury in the 65th minute after falling to the ground without any contact.

Messi was inconsolable on the bench. Floods of tears ran down his face as his ankle was strapped with ice.

His tears of sadness turned to joy by the end of extra time as Martinez's goal gifted Argentina the Copa America for a record 16th time.

However, the drama started well before kick-off, with the final being delayed by more than an hour due to fans without tickets forcibly entering the stadium.

Security staff at the stadium were forced to close the gates to the venue in an effort to regain control of the massive crowds.

"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 stadium spokesperson said in a statement. 

"Security has shut the gates in order to control the entry process at a much slower rate and ensure everyone is kept safe."

The tournament was marred by ugly scenes following Colombia's semifinal win over Uruguay, when Uruguayan players and Colombian fans began fighting in the stands. 

The final in Miami was scheduled to begin at 10am AEST, but did not kick off until about 11:20am.

Media reports have said many fans without tickets were left stranded outside as the game began.

Footage shows thousands of people rushing through the gates and climbing over fences.