The Gunners suffered a 2-1 defeat away to Manchester United at Old Trafford on Saturday, a result which left them nine points behind the table-topping Red Devils.
United's first goal was scored by Robin van Persie, who left Arsenal in the summer after announcing he would not sign a new deal with the north London club.
His departure followed those of Cesc Fabregas, Samir Nasri and Gael Clichy, and Adams told BBC Radio Five Live's Sportsweek programme: "I never got anywhere near the last year of my contract.
"You don't let them go into their last year and run their contract down.
"It's about winning, and how do you win? You have the best players, and if you keep selling your best players you're not going to win.
"If you sell your best players you're not going to challenge for titles. You need to keep your best players and in recent years we've let them go a bit easy."
Adams believes Champions League qualification via a top-four finish is the absolute best that Arsene Wenger's men can hope for.
"If Wenger gets them into the top four he's done brilliantly," he added.