The post was made after Welbeck netted the winning goal during an FA Cup quarter-final clash against his former club Manchester United on 9 March.
The slur was one of a number of messages posted in the aftermath of Welbeck’s return.
Tweeted by user @angeIsleftfoot, the offending message was posted after Welbeck’s decisive role in Arsenal’s 2-1 Quarter Final victory at Old Trafford.
Officers in Manchester with specialist training in social media identified the user and passed on the information to Wiltshire Police, who confirmed they have now made an arrest.
Welbeck was shown the door by United boss Louis van Gaal last year, who claimed he was not good enough.
The 15-year-old boy, from the Salisbury area, was arrested on suspicion of racial abuse and released on bail until 13 April, Wiltshire Police said.