The Dutchman had been expected to leave the Emirates Stadium after refusing to sign a new contract, which had been set to expire next summer.
Arsenal haven’t picked up a trophy since 2005, which seen several high-profile players depart as a result.
And Adams believes that the recent sale of Van Persie will further hinder the club’s search for silverware.
"I thought [selling] Robin [van Persie] was a mistake," Adams told BBC Sport.
"I think he could have played for four or five more years.
"I hope they do go on and win everything. Realistically, I think they will fall short again. I wouldn't have made that decision [to sell Van Persie].”
Adams also believes that former teammate Steve Bould may have problems in his new role as Arsene Wenger’s assistant.
"It's a blessing and a curse for Steve [Bould]. People are going to say 'what a fantastic defence' because he played defence," he added.
"And a goal goes in they're all going to be turning round to him and saying 'come on Steve you're doing a really bad job'.
"My experience of Arsene Wenger is that he's very offensive and it's going to be very difficult for Steve."
