Hertfordshire Police said officers were still seeking witnesses to the attack on Laird, who was knocked to the ground after the club's 3-1 FA Cup third-round defeat of Newcastle.
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.
Police and the Football Association both began investigations after Laird was floored when fans ran on to the pitch as League Two Stevenage celebrated Saturday's giant-killing.
Film of the incident was immediately circulated on the internet, showing the defender being struck in the face.
In a statement issued on Sunday, Stevenage chairman Phil Wallace said the club would do everything it could to bring the assailant to justice.