Operations Director Stuart Brown said the incident at half-time was "disappointing in the extreme".
He said he had spoken with the Rangers Head of Security and assured that the Ibrox club is fully aware of the incident and that it had been reported to the police.
Mr Brown said Strathclyde Police were treating the matter as a criminal offence.
Rangers, he said, had assured him that anyone found guilty of such an offence will be banned for life from attending matches at Ibrox.
Writing on the Stirling Albion website, he added: "In the meantime, I’ve been in contact with the young fan and his father to update them on developments and to offer my apologies on behalf of SAFC that he should be subjected to such a vile act, perpetrated by a mindless idiot for whom I hold nothing but contempt.
"As ever in football, it is disappointing that a small minority let down the great majority."