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"Most people go to watch soccer as a source of entertainment with their families, and they need to be protected from these thugs," Nematandani said in a statement.
"SAFA condemns this unruly behaviour unconditionally."
He said soccer fans should learn to respect the outcome of any game.
"We cannot allow people to take the law into their own hands, because by so doing they are destroying the 2010 (World Cup) legacy."
Nematandani's comments followed unruly behaviour from Mamelodi Sundowns supporters after their 2-0 Premiership defeat to Moroka Swallows at Dobsonville Stadium on Sunday.
Supporters tore down barriers and knocked over the club dugout in protest as their team slumped to a fourth defeat in six games.
Overzealous Sundowns supporters had also caused disturbances following the Brazilians' home-defeat to Maritzburg United last week, baying for coach Johan Neeskens' blood.
The SAFA president urged clubs and the Premier Soccer League to root out the problem through educating supporters and harshly punishing perpetrators.
"These unpleasant images are viewed across the globe," he said.
"They are denting the image of football in the country, hence we say no to hooliganism.
"The league should go through all available footage and ban for life those involved in such dangerous and unruly behaviour.
"We should root out this evil before it becomes a monster."
