Cardiff won the Welsh Cup clash 2-1, Will Fish scored the winner after Wrexham's Keiffer Moore had canceled out the Bluebirds' opener from Yousef Salech.
Afterwards, a pleased Barry-Murphy said: "Our players believe we can get to that level for sure, and that's our intention this season.
"To put ourselves in a game like this against a team who have been so impressive recently was a real test for us because you never know how it's going to go, so to perform in the manner that we did gives us great hope for the future."
He added, "We've got a very demanding owner who is very insistent that we have more shots, so hopefully he'll be very pleased tonight."
Asked if he would have a happier phone call with the 73-year-old than last time around, Barry-Murphy said wryly: "Fingers crossed."