Posting on his Facebook page, Strache wrote: "Madness in Graz! The culprit is from Bosnia. A religiously motivated assassination is not ruled out!"
The comment later disappeared from his social media page but screengrabs taken of the post quickly began circulating online.
Police ruled out a terrorist motive for the attack by the 26-year-old man who killed three people, including a four-year-old boy, when he drove his car into a crowd of shoppers on Saturday afternoon.
Regional police chief Josef Klamminger said: "We can say clearly and without doubt that it is an isolated incident without political or extremist motivation."
Initial interviews carried out by the police suggested the driver showed symptoms of "psychosis" though they say he has since remained "evasive" in his answers. He is being brought before a judge today.
Strache's post was met by fiery criticism on social media, including a comment from Austrian-Serbian football star Marko Stanković.
Stanković, who plays for SK Sturm Graz, wrote on the FPÖ leader's page: "Can you determine from these lines that the reason for this terrible act is his country of origin??? I feel personally attacked by this post and see in your post a fundamental dislike of people with foreign origin!!! Shame on you, Mr. Strache, shame on you!!!"
Strache later defended his comment by pointing to a report in the Krone Zeitung that also highlighted the driver's Bosnian origins. The driver is believed to have been born in Bosnia but moved to Austria when he was a young boy to escape the Bosnian war.