Players and officials of both teams were involved in the skirmish, which took police and security staff several minutes to defuse.
After 120 minutes the game was goalless, leading to the third penalty shoot-out of the tournament.
The crucial moment came when Justo Villar saved Franklin Lucena's fourth penalty after Paraguay had scored their first four.
Defender Dario Veron then completed Los Guaranies' 100% shoot-out display to send Paraguay through to Sunday's final against Uruguay at 20.00 BST.
Paraguay drew all three of their group games and are yet to score a goal in the knockout stages, seeing off holders Brazil on penalties after a 0-0 draw in the quarter-finals.