The England midfielder -- who scored a delightful goal to put Spurs 2-0 up -- was hit on the head as he took a throw-in on 73 minutes.
Arsenal said in a statement on Thursday the whole club had "been embarrassed by the incident" and "behaviour of this type has no place at Emirates Stadium".
The Gunners had looked at the CCTV footage and had ascertained the offender had left after the incident and they had "identified an image of the suspect".
Arsenal added once they had confirmed the identity of the fan they would pass on the offender's details to police so they could initiate legal proceedings and added the supporter faces a lengthy club ban.
"We are liaising with the Metropolitan Police and investigations continue in order to apprehend the culprit," read the Arsenal statement.
Spurs manager Mauricio Pochettino praised Alli's reaction to the incident in rubbing the back of his head and gesturing towards the Arsenal fans to indicate the scoreline.
"It was an amazing reaction," Pochettino said at his press conference on Thursday.
"In another country maybe the player would create a problem. Dele was so honest and professional and he behaved really well.
"It was top behaviour.
"Sometimes people criticise Dele for his actions on and off the pitch but it's good to praise him today."
The incident came after a Tottenham fan was banned for four years after being found guilty of throwing a banana skin at Arsenal's Pierre-Emerick Aubameyang when the teams met in an acrimonious clash earlier this month.