The Blues currently lie 22nd in League Two but Barry believes the performance in Saturday's draw with Grimsby could prove a turning point.
"The lads know we can stay in the Football League and we're all eager to prove we can do a job here at Chester," he told the Evening Leader.
"Everybody was pleased that we managed to produce a much improved performance on Saturday and come away with a point in our home draw against Grimsby.
"We're now determined to push on and build on that by getting results from our next two games at Bury and then at home to Bradford," he added.
"I still think we can turn things round even if games are now starting to run out."