The Addicks were booed off the pitch after Scott McDonald's deflected strike settled a low-key derby encounter.
Chris Powell's side have slumped to 20th in the table and face a tough trip to in-form Burnley next weekend, but right-back Wilson is confident they can bounce back.
"We'd like to apologise for the loss," he said.
"It hurts us just as much. Fans want to see us win and if we're not winning, we're not happy and the fans are not happy.
"They're entitled to boo but the boys will come back and hopefully get a win out of next week."