Former Labour minister Ms Cooper accused Mr Johnson of having a "shameful lack of judgement" and a willingness to play "the most divisive, cynical politics".
She added: "He should not try to play political games with the darkest and most serious chapter of Europe's history. The EU has played a critical role keeping peace in Europe ever since."
'Vital projects' jeopardised
The exchange comes as the campaign on both sides of the debate stepped up a gear on Saturday.
With less than six weeks to go until the vote, polls have put the Remain and Leave campaigns at roughly 50-50.
Mr Cameron warned that leaving the EU would have a "devastating impact" on infrastructure projects.
Campaigning in his Oxfordshire constituency, he said "vital projects across every region of the UK" would lose "crucial funding".
Mr Cameron said projects that had benefited from such funding included new trains for the East Coast Main Line, the M8 motorway extension between Edinburgh and Glasgow and the expansion of research and teaching facilities at Oxford University.
Meanwhile, Labour leader Jeremy Corbyn encouraged his supporters at a rally in London to vote Remain, saying it was "not the European Union that's the problem - it's the Conservative government".
Campaigning in Bristol, Mr Johnson said the UK should leave because the EU has changed "out of all recognition from what we signed up for".
He said the UK's "aborted" negotiations with the EU showed it was "going in completely the wrong direction".
"All our attempts at reform are hopeless unless we vote to leave," he added.
Grassroots Out, which is also calling for the UK to leave the EU, held a series of nationwide events on Saturday.
Meanwhile, the official Britain Stronger In Europe campaign said put on about 1,000 events across the UK.