Mugabe's resignation followed an impeachment motion adopted by Zimbabwe's parliament on Tuesday.
"As Zimbabwe's oldest friend we will do all we can to support this, working with our global and regional partners to help the country achieve the brighter future it so deserves".
"The DA has today written to President Jacob Zuma and relevant ministers to put them on terms to not entertain granting President of Zimbabwe, Robert Mugabe, or his wife, Grace Mugabe, political asylum in South Africa".
"It reflects the importance of Zimbabwe psychologically to the United Kingdom, and the government attempt to show leadership among Western governments in how to respond to the leadership change there", he said.
Lawmakers roared in jubilation and people have begun celebrating in the streets.
Mugabe's successor, former Vice President Emmerson Mnangagwa, will be inaugurated on Friday. This should assist with the peaceful transition to a new leadership.
On Thursday, global news agency Reuters quoted a senior International Monetary Fund (IMF) official as saying Zimbabwe's economic growth was threatened by high government spending, an untenable foreign exchange regime and inadequate reforms.
That was after Zanu PF had booted as its leader on Sunday.
Zimbabwean advocacy group, the Election Resource Centre (ERC), on Wednesday welcomed Mugabe's resignation but highlighted the efforts Zimbabweans must make to ensure that a democratic election.
"And yet, tragically, he allowed that legacy to be squandered and his country went to rack and ruin and in some cases his people were driven to the brink of starvation".
However, Mugabe and his family own several properties in South Africa and the couple's two sons, Robert Jnr and Chatunga live there.