Has a Copa America game ever ended 5-4?
Yes. Copa America has seen a final score of 5-4 4 times. It first happened on March 6, 1955, when Chile beat Peru, and most recently on March 31, 1963, when Bolivia beat Brazil.
That makes 5-4 the 23rd most common final score out of 36 that have ever occurred in Copa America. One point away, 6-4 has happened 2 times and 5-5 has happened never. Nearly all of its occurrences fall between 1950 and 1979.