Ludlow volleyed home from close range following a corner from England international Karen Carney, after a tight match.
The crowd was the highest ever for a FA Women's Premier League Cup final, with Iron fans who attended able to see some of the stars from the England national side - who will compete in the World Cup later this year - in action.
Arsenal are in line for a record-breaking season - they top the Premier League, are in the final of the UEFA Women's Cup and have aspirations of lifting the FA Women's Cup again.
