Gourlay left Stamford Bridge last week and Chelsea have confirmed chairman Bruce Buck and director Marina Granovskaia will assume the roles formerly held by him.
Purslow - formerly managing director at Premier League side Liverpool - will join the club immediately.
"The club has ambitious plans to build the Premier League's most pioneering global commercial programme, partnering with innovative and market-leading organisations from around the world," read a Chelsea statement.
"The club believes Mr Purslow has the vision and the leadership qualities to help us achieve these plans.
"Chelsea does not expect to announce any other senior appointments in the near future. As announced last week, chairman Bruce Buck and director Marina Granovskaia have assumed the executive responsibilities held by our former chief executive."
