The deal to buyout 50% partner Lloyds Banking Group was concluded on January 31, giving Sainsbury's full autonomy to develop the business.
Six months ago the bank signed Edinburgh's biggest private sector office deal in a decade and forecast a 20% rise in the workforce in 2013.
Now Mr Griffiths has said the 320 headcount in the capital will grow by up to 80 during the current year.
The retail giant has 23 million shopper visits every week and 1.5m active bank customers.
Mr Griffiths said: "The challenge over the next two years is to get our own banking platform up and built, and migrate customers across to the new bank. It gives us far more flexibility to extend the product range and launch new things."
But he said Sainsbury's had no intention of following Tesco into current accounts, which its rival is now piloting. "They are a relationship product, I would argue we already have a relationship with the Sainsbury's shopper."
The bank offers extra Nectar points and shopping discounts to customers taking out more financial products.
Mr Griffiths, former chairman of the Building Societies Association, said: "We have had five strong years of profit growth and over the last four years probably made more money than a lot of the big four banks.
"It is a prudent business, we know what we are trying to do with it and we will invest very heavily in Edinburgh."