The present invention is directed to a flex pack wipes dispensers for dispensing wipes. An exemplary wipes dispenser may include a flexible container body having an opening through which the wipes are dispensed. A plurality of wipes (e.g., a stack) is contained in the flexible container body. A closure is provided over the opening through the flexible container body, configured to allow a user to selectively open and close access to the opening. The closure may include a perimeter flange sealed to the flexible container body, a base defined within the flange which initially covers the opening in the flexible container body, and a tear-away orifice covering in the base. A shape of the covering includes a perimeter following a pathway that includes a portion forming an angle that is greater than 90° relative to a direction of pull used to tear the covering from the base.