Hertfordshire Police said officers were still seeking witnesses to the attack on Laird, even though footage of the incident has been shown on U-Tube.
A police spokeswoman said a 24-year-old man from Stevenage was arrested in connection with the incident after handing himself in at a local police station on Monday.
In a statement issued yesterday, Stevenage chairman Phil Wallace said the club would do everything it could to bring the assailant to justice.
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Stevenage chairman Phil Wallace has launched an investigation into the attack on defender Scott Laird.
Phil Wallace: "I wasn't aware of the assault on Scott until long after the game.
"When I was told I spent a considerable time viewing the incident from all angles with representatives of the police.
"The assailant can be seen clearly from the moment he enters the pitch.
"He has approached at least three Stevenage players, seeming to wait for an opportunity, and thrown a punch to the side of the head that was designed to inflict maximum damage.
"Scotty collapsed unconscious and has no recollection of the incident.
"This person seemed intent on making a random assault on a Stevenage player after a fantastic victory.
"This is disgraceful behaviour and we will do everything we can to ensure that the assailant is brought to justice and prosecuted.
"Recognition should not be difficult as the video images of him are crystal clear.
"This time it was a fist, who knows what it could be next time?
"We do everything we can as a club to prevent this type of incident and follow all the recommended procedures.
"But it seems this lad has set out to disguise his intent so that he can cause physical harm to one of our players and we will work with police tirelessly until this person is identified and prosecuted."
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Police and the Football Association launched an investigation after a Stevenage player was punched, apparently by one of his own fans, after the shock FA Cup victory over Newcastle.
Defender Scott Laird was floored by the blow as fans ran on to the pitch at the final whistle as the League Two side celebrated yesterday's 3-1 win over the Toon.
At first it was thought to be a Newcastle fan ... but if you look close to the video footage he is carrying a Stevenage scarf.
Laird was celebrating on the halfway line following the game, then he is suddenly approached by the fan who punches Laird directly in the face, knocking the player to the turf.
Stevenage manager Graham Westley: "Scott Laird was stood celebrating with the fans and a fan just came over and smashed him from the side.
"He went to the ground and it is a shame for the afternoon to end in that way.
"He is a bit bewildered. He (the fan) gave him a right-hander, just smashed him across the cheek and knocked him to the ground.
"He said to me, 'It's lucky I've got a big head to take it'. It was a sad way for things to end."
An FA spokesman said: "It's something that should be flagged up in our meeting on Monday.
"It might be that there was a stadium adviser at the game who'll be able to advise us on that."
Police have described the fan as a white male wearing a white Diesel top and red and white Stevenage scarf.
LOOK AT THE FOOTAGE SENT TO US BY A STEVENAGE FAN FROM HIS MOBILE PHONE:
Watch from 1:36, as he jumps from the Stevenage fans, wearing similar clothing to the clip showing the punch and waving a scarf in right hand.
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