Celtic's 3-0 victory came on a day when both teams showed their support for Show Racism the Red Card's annual fortnight of action.
Celtic's Cha Du-Ri has accused St Johnstone supporters of making monkey noises at South Korean team-mate Ki Sung Yung. Du-Ri said: "When Ki had the ball, two supporters jumped up and started making monkey noises in unison.
"I played for eight years in Germany and I have never seen anything like that.
"The incident is shameful and I feel angry about it. I told the coach this when we had dinner after the game."
A statement on the club's official website said: "It has been brought to the club's attention that an extremely small minority of our support may have been responsible for making racist noises towards certain Celtic players during Saturday's game.
"St Johnstone FC will conduct a thorough investigation into these allegations. We will consult both the police and our stewarding suppliers, G4S.
"Additionally, we are appealing to fair-minded Saints fans, who make up the overwhelming majority of our support, to let us know if they are able to identify the individuals who may have been responsible.
"Should we be able to identify any perpetrators, the most stringent action will be considered."