Speaking on the BBC's Working Lunch TV programme, Leighton said: "What's happening in the country now is really going to be powerful for the Post Office. Post offices offer good old banking basics, and suddenly boring, basic banking is one of the safest places to have your money."
Thousands of people had put money into PO savings accounts in recent months, he said.
Leighton was responding to complaints by retailer David Ward of East Barkwith, Lincolnshire, who was angry about post office closures and the removal of traditional duties from remaining outlets.
Wearing a My Shop is Your Shop 'Local and Proud of it'
T-shirt, David said: "It's frustrating that the government is giving PO business away to private contractors - it just seems crazy. Without the post office, my shop would just not be viable."
Leighton defended the closures. "When we close an outlet, 80% of the business goes to another post office and makes it stronger," he said. "We have the right number of outlets now and are pretty close to a vibrant, viable post office community."