The Labour leader said Whitehall and councils could obtain new opportunities to use their “incredible” purchasing power to support UK businesses and boost local economies.
Labour also wants firms bidding for public contracts to have to recognise unions and curb “boardroom excess”.
On a visit to Doncaster, Mr Corbyn said Brexit presented “many challenges”.
But the Labour leader, who campaigned to remain in the EU, added that severing ties with Brussels could also “give us more powers to encourage best practices and support new and existing businesses and industries in Britain”.
Labour said that because EU rules meant certain public contracts currently had to be opened to competition from elsewhere in Europe, Brexit could “allow public bodies to use local pounds on local jobs and businesses”.
It said this would “therefore create scope for additional flexibility” for public bodies to “require the use of local or regional suppliers after Brexit”, meaning cash could be directed “back into local economies”.
Mr Corbyn added: “While the Conservatives seem intent on using Brexit to turn us into a low-wage tax haven, Labour will use every power possible to upgrade our economy so we can all lead richer lives.”
The UK is set to leave the EU in March 2019.
Labour also said the UK should assess the impact on local councils of its membership of a World Trade Organization procurement agreement – which the UK is part of as an EU member – requiring contracts above certain thresholds to be opened up to foreign competition.
Labour has previously said that firms bidding for contracts would need to comply with collective bargaining agreements, pay suppliers within 30 days, maintain high environmental standards, provide training and apprenticeship opportunities, comply fully with tax requirements and adopt “best practices” in equal opportunities.
All of this is possible within existing EU rules, it said.
Conservative Party chairman Patrick McLoughlin attacked Labour’s bid to force companies bidding for government contracts to recognise trade unions.
“This cynical attempt by the Labour Party to give more power to their union paymasters would force up council procurement costs, causing council tax to soar,” he said.