Anonymous Africa said it had also hacked into the sites of Zimbabwe’s defence ministry and the state-run Herald newspaper.
The ANC said its site had been “flooded” by a Distributed Denial of Service (DDS), which overwhelms sites with huge amounts of traffic.
Mr Mugabe has called elections in July.
South Africa’s President Jacob Zuma is the lead mediator between Zimbabwe’s leader and his long-time rival, Prime Minister Morgan Tsvangirai.
At 12:30 BST (11:30 GMT) all three websites were working again.
ANC spokesman Jackson Mthembu said the party’s website team was “currently working on the problem, including assessing means to strengthen our security so that such does not recur in future”.
On its Twitter feed @zim4thewin, Anonymous Africa said the ANC was an “enabler” of Mr Mugabe’s government.
It said it had targeted the Zimbabwe defence ministry because of the “genocide” of 20,000 Ndebele people in the 1980s.
Zimbabwe’s government has always denied accusations that it deliberately killed civilians because they were ethnic Ndebeles, seen as supporters of Mr Mugabe’s rival, the late Joshua Nkomo.
It says it was targeting criminals.
Some of Mr Tsvangirai’s supporters want South Africa to put more pressure on Mr Mugabe to ensure that elections are free and fair.
On Thursday, Mr Tsvangirai accused Mr Mugabe of acting “unlawfully” by saying elections would be held on 31 July.
The prime minister wants more time to ensure that reforms are in place before polling day.
Mr Mugabe says he is complying with a court order, which said the elections must be held by the end of July.