Rowing, 70, announced he was stepping down from the role on Monday and sports lawyer Mansford will work alongside new club chairman Maurice Watkins in a new-look boardroom intent on improving the club's fortunes both on and off the pitch.
Mansford, leaving his current role as director of Los Angeles-based sports marketing and management firm Wasserman Media Group, told the club's official website: "I hope to be in post very soon and to meeting the fans and stakeholders. We are one town, one club and one team.
"We need Oakwell to become the place where people come to watch football and enjoy themselves during what are still tough economic times.
"It is a great honour and a privilege to be involved in the management of such a historic club.
"I'm looking forward to the challenge of continuing to show that anything is possible.
"Barnsley FC is in good shape thanks to a good job by Don Rowing off the pitch and in recent times on the pitch as David Flitcroft has driven the players forward in remarkable fashion."
Watkins added: "In Ben we have someone who not only understands the management and finances of football and has a good record of business achievement in sport, but also shares our deep desire for Barnsley FC to succeed.
"I'm sure he will prove an excellent chief executive and can build on Don Rowing's legacy of continuous improvement."
Mansford, who lives in East Yorkshire, worked as a sports solicitor with Leeds-based Walker Morris prior to joining WMG.