Zenit trailed Dinamo 4-2 in an encounter that Villas-Boas' side needed to win to ensure they would head into the final weekend of the season top of the table in a three-way title race with CSKA Moscow and Lokomotiv Moscow.
However, the home fans at the Petrovsky Stadium charged into the area behind the away goal in the 87th minute of the contest.
"It's really sad that this situation took place in our match," Villas-Boas said. "Things like that shouldn't happen.
"It's not acceptable for fans to come on the pitch during the match, just as it's not acceptable for them to be aggressive towards players.
"There can be no violence or racism in football. Of course, clubs, and particularly our club have to implement serious and smart work with fans.
"Of course, we are awaiting the decision by the CDC (Control and Disciplinary Committee).
"Unfortunately, the stadium can be punished, and in this case, the team will play either without spectators, or at another stadium, and that's bad, because fan support is very important to us."
