Mosimane was subjected to vile insults and behaviour from a section of Mamelodi Sundowns supporters in Attridgeville on Saturday before their second-leg win over his former club.
After the 3-1 aggregate scoreline which saw the Red Devils advance to the semi-finals, where they will face Wydad Casablanca, he was visibly emotional about the incident and revealed they even insulted his mother.
"These individuals have not only humiliated coach Pitso Mosimane, they have brought the name of the country into disrepute and as SAFA we will leave no stone unturned in identifying these individuals and handing out the harshest of penalties to the perpetrators," SAFA CEO Advocate Tebogo Motlanthe said.
"We have requested for an urgent report from the security and police at the stadium and we will also await a report from the club. We are treating this matter with the urgency and seriousness it deserves. We cannot tolerate this; Mosimane is our proud ambassador and what these people did needs to be condemned by all civilised society."
The Economic Freedom Fighters (EFF) also condemned the unsavoury scenes and backed the two-time CAF Champions League-winning coach for his contributions to African football.
"Sundowns has shown the highest form of self-hate horrendous to watch; worse Pitso Mosimane's mother was dragged into a verbal assault that was obviously premeditated to bring Mosimane," read part of the statement.
"He built Sundowns and won all possible trophies‚ and if anything‚ Sundowns should be celebrating him."
The Brazilians released an apology statement to Al Ahly on Sunday in the aftermath of the intense backlash on social media following Mosimane's post-match press conference.