As well as the three killings, Rudakubana pleaded guilty to the attempted murder of eight other children and two adults injured in the attack.
Searches of his home uncovered material which suggested an obsession with violence, including the academic study of an al-Qaeda training manual downloaded from the internet.
Police have said Rudakubana used techniques set out in the manual when he carried out the attack.
In the letter, which was co-written with Technology Secretary Peter Kyle, Cooper said that, while possession of the document was illegal under existing anti-terror laws, Rudakubana had been “able to easily obtain access” to it and that it “continues to remain available online”.
“The ease of access to such dangerous, illegal content is unacceptable,” she said.
“We must prevent further unlawful access to instructional material by those wishing to do harm to the United Kingdom.”
Before leaving home to carry out the attack, Rudakubana watched graphic footage of a knife attack on Australian bishop Mari Emmanuel that had taken place a few months earlier.
Cooper noted the video of the attack had been removed in Australia but could still be accessed in the UK, adding the Southport attack had “laid bare the potential consequences of failing to act on such content”.
“We are therefore urgently asking you to ensure this is also removed from your platforms to ensure that it is not used to inspire other attacks,” she said.
From March, under the Online Safety Act, which passed in October, social media platforms will be obliged to to take “proportionate measures” to protect users from illegal content.
Cooper said there was “no justification for waiting for laws to kick in” and that platforms had a “moral responsibility to act now”.
