Kirkland, who was playing for Sheffield Wednesday, was knocked down after the visitors had equalised and needed medical attention before he could continue. The shocking scenes were broadcast live on television.
Outraged fans took to Twitter and Facebook to name and shame a man suspected of the attack on Kirkland.
South Yorkshire Police's senior investigating officer, Detective Inspector Mark Monteiro, said: "With the public's assistance we have clearly identified the offender who ran on to the pitch during last night's game between Sheffield Wednesday FC and Leeds United FC and who is suspected of assaulting the Sheffield Wednesday goalkeeper.
"We are dealing with the complaint made by the player and we are currently taking statements, collating evidence and making inquiries to locate and arrest the offender.
"We are also using CCTV footage from around the ground to identify suspects who may have been responsible for other offences before, during or after the match."
There were five arrests for various offences including public order before and after the game, while three people were ejected from the ground and 12 were subject to dispersal orders.
An FA spokesman said: "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."
The match was marred by vicious chanting from some Leeds fans about Wednesday manager Dave Jones, who was cleared of child abuse allegations in 2000.
Wednesday fans replied with songs about two Leeds fans killed in Turkey in April 2000. Leeds supporters responded by throwing bottles on to the field and clashing with police.
Leeds issued a statement after the game pledging to identify the man involved in the incident.
It read: "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."
Jones called for Leeds fans to be banned from "every away ground".
He went on: "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."