Justin King said consumers are "completely in the dark" over the effect leaving the EU will have on their shopping basket.
Speaking to BBC Panorama, he said it was "very clear" shoppers would face "higher prices, less choice and poorer quality" outside the bloc.
"Brexit, almost in whatever version it is, will introduce barriers," he said.
"That makes it less efficient which means all three of those benefits - prices, quality and choice - go backwards."
Mr King, who was chief executive of Sainsbury's between 2004 and 2014, said putting up prices was the "last thing" any current supermarket boss would reveal.
However a monthly tracker showed the cost of an average basket of groceries rose sharply in June.
A basket of 35 popular items came to £83.04 - £1.67 more expensive than in May and 21p higher than this time last year, according to mySupermarket's monthly Groceries Tracker found.
The figures brought to an end a continued fall in the price of the basket amid warnings of post-Brexit increases to the cost of imported goods.
A Department for Environment, Food and Rural Affairs spokeswoman said: "As the Secretary of State has made absolutely clear, there will be no diminution or watering down of food standards.
"Leaving the EU provides us with a golden opportunity to develop a new farming and food policy. We will remain global leaders in environmental and animal welfare standards, maintain our high quality produce abroad and reduce pressure on the weekly household budget."