An exhaust connection coupler and a method for manufacturing a braid cover incorporated therein are provided. The method of manufacturing comprises the steps of providing a tubular sleeve formed of braided wire filaments and forming at least one end of the sleeve into shape using a forming die such that the at least one end of the cover includes a circumferential neck portion having a diameter smaller than the diameter of a body portion of the cover. In the method, the sleeve may be pressed between male and female forming dies in order to create the cover's desired shape, which can include a shoulder extending radially outwardly from the cover's neck portion and meeting the cover's body portion at a corner.