Chief Superintendent Jason Harwin, of South Yorkshire Police, said a senior investigating officer would be working with a team of officers to identify those responsible for the disruption.
An FA spokesman confirmed on Saturday that an investigation is already under way at football's governing body.
"We are investigating the events and awaiting the referee's report," a spokesman said.
"The FA condemns the scenes at Hillsborough. 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."
Kirkland, a 31-year-old former Wigan Athletic keeper who picked up one England cap in 2006, was knocked to the floor with a violent two-handed shove to the face after a fan ran on the pitch following the Leeds equaliser in the 77th minute.
The Leeds United fan could clearly be seen smiling as he ran back into the ground while the dazed Kirkland lay prostrate. After medical treatment, Kirkland was able to continue for the remainder of the Yorkshire derby clash.
The Leeds fans amongst the crowd of 28,582 also courted controversy by singing songs that referenced criminal charges against Wednesday manager Dave Jones which were eventually dropped, as well as seeming to salute Jimmy Savile, the former broadcaster who is now the subject of a criminal investigation for child abuse.
An unhappy Jones called on Leeds United fans to be banned from future away matches after branding them "vile animals".
Leeds United were quick to release a statement denouncing the incident.
"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."
The sentiments were shared by Leeds manager Neil Warnock. "I'm absolutely embarrassed - absolute disgrace," said Warnock, who also seemed to suggest Kirkland had exaggerated his fall.
"Listen, he (Kirkland) went down like a ton of bricks, but that doesn't make a difference," said Warnock. "There's no way he should be on the pitch doing that, I hope we can make an absolute issue of it.
"It spoilt everything for everybody. When I see a moron like that I'm not proud to be Leeds manager. I don't mind the rivalry - elbows flying, penalties missed by the referee - but to see things like that...I'm absolutely embarrassed.
"They had enough security men, but if you've got a person like him with what must be a five-year-old's mentality then you can't stop it. Get the guy, prosecute him and put him in bloody prison. Our crowd was fantastic tonight - we had one moron. I'm very sorry about that."
Kirkland had just conceded a 77th-minute equaliser to Michael Tonge when the incident happened, with a man running onto the field from the Leppings Lane End which was housing the Leeds fans, and appearing to punch Kirkland in the face.
The man then returned to the stand while trouble continued around him, with a number of other visiting supporters running on and off the field and stewards struggled to contain them.
Tonge, who spent a large chunk of his career under former Sheffield United boss Warnock, fired home a swerving 25-yard shot in the 76th minute to cancel out Jay Bothroyd's first-half header and deny Wednesday a first win in eight league matches.
But despite rescuing a point with a much-improved second-half display, Leeds are now sure to find themselves in serious trouble from the game's authorities.
Riot police had earlier filtered out to try and diffuse ugly scenes in the away end prior to Tonge's strike, with bottles thrown between baying sets of home and away supporters.
Leeds boss Warnock, returning to his home city for the first time since former club Crystal Palace triumphed over the Owls at Hillsborough in an FA Cup tie in 2010, will have been grateful for Tonge's rescue act after his side stretched their unbeaten run to five matches despite being outplayed for long periods.
Leeds were given a real let-off in the 23rd minute by referee Eddie Ilderton, who somehow failed to see Leeds forward Luciano Becchio handle the ball when diving in his own penalty area to head clear Bothroyd's 23rd-minute corner.
But Wednesday, belying their lowly status, secured the lead they deserved two minutes before the break. Long-throw specialist Michail Antonio launched a missile into the Leeds box from the left and Bothroyd rose highest to head home unopposed, low into Kenny's right-hand corner.
Leeds were more purposeful after the restart and went close to an equaliser through another free-kick from Tongue after Gardner had been booked for his high tackle on Diouf.
Tonge's brilliant equaliser in the 76th minute following Aidy White's lay-off then prompted the disgraceful scenes as some Leeds fans to pour through fencing.
Kirkland required treatment before the game could resume and although Owls substitute Danny Mayor went close with a shot blocked by Jason Pearce in the closing stages, the home side had to settle for a point.
- Leeds United