Marsh will join DS Smith from Tesco where he was the group director of tax and corporate finance for three and a half years.
He was also chief financial officer (CFO) at AstraZeneca and CFO of Global Building Products at Pilkington.
Dryden, who announced his retirement in January, will stand down from the board and leave the company on 28 June, following publication of the company's final results.
David Matthews, group financial controller, will be interim group finance director until Marsh takes over in September.
DS Smith chairman Gareth Davis said: "I am delighted to announce the appointment of Adrian as group finance director. He brings extensive and valuable international experience and skills and I am confident he will make a great contribution to DS Smith as we continue to execute our strategy to become the leader in recycled packaging for consumer goods."
Marsh added: "I am very excited to be joining DS Smith at such an important stage in the company's history. I have been extremely impressed by the business and the management and look forward to supporting the Board in driving the company's development."
