The ex-Labour leader said he was in favour of devolving powers to the Scottish Parliament to "deal with these issues".
But he hit out at both the SNP and the Tories, accusing them of playing a "blame game" about who was responsible for the problem.
While Conservatives have challenged the Scottish Government to do more to address the drugs problem, SNP ministers have complained that they do not have powers needed to set up drugs consumption rooms.
Mr Brown said: "I'm in favour of making the powers available to the Scottish Parliament to deal with these issues."
However speaking at an event organised by the Our Scottish Future think tank that he founded to "build the positive, progressive and patriotic case for a better Scotland in a reformed UK", he also stressed the need for greater co-operation.
"The borders do not matter to the drugs traders so we have got to find a way of working together," Mr Brown said.
"Let's get us working together so we can deal with this terrible blight of drugs deaths which is really disfiguring our country and causing so much harm to families."
He insisted: "Whatever happens you have got to cooperate on drugs, because the drugs come from all sorts of parts of the world where you need police forces to work together to control them. You need cooperation.
"We have got human lives at stake, how awful it is to have a blame game when you should be dealing with the problem."
His comments came after a speech in which he spoke about the need for cooperation across the UK "to make the best of our country and make nationalism, that divides the worlds into warring tribes, a relic of the past".
He told the audience: "We must not let nationalists, the destroyers of Britain, ignore what Scotland and Scottish people agree on more than anything else - 75%, three-quarters, of all Scottish people in every poll we do are in favour of more cooperation not less with the rest of the United Kingdom."
Mr Brown, who played a key part in the campaign to keep Scotland in the UK in the run-up to the 2014 independence referendum, added that polling for Our Scottish Future had found 65% of Scots want more cooperation with the rest of the UK on health issues such as dealing with pandemics, that 66% want more cooperation on the economy and 61% support greater cooperation on tackling climate change.
Earlier this year, Mr Brown criticised current Prime Minister Boris Johnson, and accused him of seeking to “bypass co-operation and impose his view on Scotland".
Responding to the Prime Minister’s remark that “devolution is a disaster”, he said that he believed Mr Johnson felt that “we should really just be ruled by him from London”.