Mr Barclay made the claim about the potential for the region's businesses during a visit to Abingdon Health, a diagnostics firm based at the National Agri-Food Innovation Campus at Sand Hutton, near York.
He said firms like Abingdon were an example of the "great opportunities we have through Brexit to trade with new markets around the world".
Mr Barclay said: "It is a business that is growing, revenue has doubled, it takes cutting-edge science in our universities and turns them into product, a bit like the pregnancy tests that people are familiar with at home.
"It takes those products, whether it is for health or animal testing, whether it is water and allows us to export that around the world.
"There are huge opportunities for trade deals in Brexit in countries like Japan with a growing elderly population and it is businesses here in Yorkshire that have the potential to take the best in our science, apply it through the manufacturing we are doing here in York and transfer that into exports around the world."
A free trade deal between the European Union and Japan was agreed last December, creating a trading zone comprising 635 million people and a third of global GDP. In June the Japanese foreign minister said trade talks could not take place with UK on a replacement trade deal until after Brexit.
Separately, Business Secretary Greg Clark has warned the next Prime Minister that a no-deal Brexit would lead to the loss of "many thousands" of jobs.
Mr Clark implored colleagues to "strain every sinew to avoid that", with leadership rivals Boris Johnson and Jeremy Hunt both pledging to leave the EU without a deal if need be.
Mr Barclay said during his visit that he was "confident that Abingdon Health, and places like it, will thrive after Brexit" and that officials were looking at working closely with Europe on "mutually beneficial areas such as research and development".
He said the UK would continue making funding contributions to maintain access to Horizon Europe, the multi-billion pound European Union research and innovation programme.
And he added: "Were the UK not to participate in Horizon in future, as a third country, it would detriment to the programme - both in terms of finance and excellence.
"The Medicines and Healthcare products Regulatory Agency (MHRA) is making necessary preparations so they can continue to regulate and export their products under existing arrangements, and the Chancellor has announced additional funding for the British Investment Bank should we lose access to the European Investment Bank.
"Leaving the EU will allow us to drive greater openness with collaboration partners in Europe and beyond - enabling companies like Abingdon Health to remain at the forefront of their field."