Kirkland was pushed two-handed in the face by a man who ran onto the pitch from the Leeds end at Hillsborough after Michael Tonge had smashed in the visitors' equaliser.
An FA spokesperson told Press Association Sport: "We are investigating the events of last night and awaiting the referee's report.
"The FA condemns the scenes at Hillsborough last night. We have begun an immediate investigation and have been in contact with both clubs.
"We would call for the authorities to identify and bring before the courts anyone who was involved in any form of criminal disorder."
Several Leeds supporters ran onto the pitch following Tonge's goal and Whites fans also flocked to a corner of the stand, ripping seats up and throwing bottles because of some reported jibes from Wednesday fans.
South Yorkshire Police also said that they were probing the incident.