Has a Copa America game ever ended 5-3?
It has. 5-3 has been a Copa America final 4 times in 869 games. The first was Argentina beat Paraguay on March 2, 1955; the latest was Paraguay beat Venezuela on February 1, 1967.
That makes 5-3 the 22nd most common final score out of 36 that have ever occurred in Copa America. One point away, 6-3 has happened 1 times and 5-4 has happened 2 times. Nearly all of its occurrences fall between 1950 and 1979.