A number of Leeds supporters ran on to the pitch after Michael Tonge equalised in the 1-1 draw at Hillsborough and one man confronted Kirkland before shoving him in the face with both hands, knocking him to the floor.
An FA spokesman: "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."
Chief Superintendent Jason Harwin, of South Yorkshire Police: "First I would like to thank members of the public who have contacted us with information to help in the ongoing investigations following the incidents at Hillsborough tonight.
"Fans have now left the city and there has been no further reported incidents of violence and disorder."
Leeds United statement: "Leeds United Football Club would like to publicly apologise and condemn the action of the fan who came on the pitch and attacked Sheffield Wednesday goalkeeper Chris Kirkland.
"The club will fully co-operate with the police and the football authorities in identifying the individual concerned.
"After the week football has endured there is no place for this type of behaviour and the majority of Leeds supporters will be ashamed of his actions."
Sheffield Wednesday manager *Dave Jones has called for Leeds fans to be banned from "every away ground".
(*In June 1999 Jones was formally questioned by police over alleged sexual abuse at St George's School in Formby, Merseyside, a home for children with educational and behavioural problems, where he had been employed as a care worker from 1986 to 1990.
After voluntarily attending the police station, he was arrested then questioned, before being released on bail without charge.
He was subsequently charged on 27 September with nine offences against young boys of indecent assault and child cruelty. He denied all the allegations and stated he was "confident that innocence will be established in due course". He appeared before Merseyside Magistrates Court on 2 November 1999 where he formally pleaded not guilty to all charges and was granted bail.
The case reached Liverpool Crown Court in December 2000, by which time Jones had parted company with Southampton. He stood trial on an eventual 21 charges, which was swiftly reduced to 14 after two other alleged victims pulled out of proceedings on the eve of the trial. After a further alleged victim declined to appear or refused to give evidence, the Judge directed the jury during the fourth day of proceedings to return a formal not guilty verdict on four charges relating to the absent party. After decreeing a retrial would not be "just" on the remaining charges, the Judge recorded not guilty verdicts on the remaining 10 charges. Jones left cleared of all allegations and was told by the Judge: "No wrongdoing whatsoever on your part has been established".)
A large number of the Leeds fans housed in the Leppings Lane End sang about the child abuse charged which were dropped against Jones in, while former broadcaster Sir Jimmy Savile, now the subject of a similar police investigation, seemed to be heralded in song.
The Wednesday fans also stoked things up with songs about the two Leeds fans killed in Turkey in April 2000, with the Leeds supporters responding by throwing bottles onto the field and clashing with police.
Dave Jones: "They are vile animals.
"You talk about racism, the chanting, that's a form of racism. And then (for the Leeds players) to go and wave at them... not in my book. You don't wave at them after that.
"I thought it had gone out of our game. The authorities have to look at it and sort it. I'm talking about an incident when we should be talking about the football. We're talking about vile animals. That's what they are. We talk about racism, but that, we need to sort that.
"It wasn't just one chanting. That was from their area too. I've had it for 12 years. It's just upsetting that they're allowed to do it. They should be banned from every away ground."