Toney, who scored a header in the 74th minute to cancel out Leandro Trossard's earlier strike, was also targeted earlier this season in October after Brentford's 2-0 win over Brighton & Hove Albion.
"Immediately after Brentford's game against Arsenal, Ivan Toney received a barrage of abusive, racist direct messages via his Instagram account," the club said on Sunday.
"We are disgusted and saddened that Ivan has had to deal with this yet again."
Arsenal said they were working with Brentford to identify those involved.
"We stand with Ivan Toney," Arsenal said. "At Arsenal, we condemn all forms of discrimination and take a zero-tolerance approach. We will apply the strongest possible action to anyone we can identify sending hateful messages."
Toney's goal cleared a lengthy VAR check but Arsenal manager Mikel Arteta later questioned the decision, saying it should have been disallowed for offside.
Referees' body PGMOL accepted on Sunday it was the wrong call and was one of two "significant errors" over weekend's games.
