The BBC's Arab affairs editor Sebastian Usher says the killing of Mr Saad is a blow to Saudi-led efforts to re-establish Aden as a secure base for the government which spent months in exile in Saudi Arabia.
Mr Saad was a significant figure not just as the administrative head of Aden, but for the role he played in driving Houthi rebels out of the port city earlier this year, our correspondent says.
But Aden has remained vulnerable to violence with jihadists carrying out regular attacks.
The claim by IS introduces another dangerous factor into the equation, our correspondent says, because like the long established al-Qaeda franchise in Yemen, IS has gained strength from the violence and chaos of the past nine months of all-out conflict.
Mr Saad's murder is also likely to complicate further the latest UN-led efforts to get a peace process under way.
IS has endeavoured to make the situation even worse, our correspondent says, by bombing mosques and killing captives in its trademark style of grotesque and horrifying showmanship.
In a statement, IS promised more operations against "the heads of apostasy in Yemen" along with photos of a booby-trapped vehicle which detonated as a white vehicle carrying Mr Saad drove past.
A witness of the blast told the BBC that the explosion was "very powerful" the ensuing fire was "very intense".
"We pulled some wounded people out but we couldn't get the governor out because [his] car was on fire," Abu Mohammed said.
Sunday's violence came after the UN envoy to Yemen met President Abdrabbuh Mansour Hadi in Aden on Saturday in an effort to bring eight months of civil war to an end.
Air strikes and fighting on the ground in Yemen have killed more than 5,700 people since the Saudi-led coalition began a campaign to restore the government in March, according to the UN.
The UN hopes to organise talks later this month between the government and the Houthi rebels, who support former President Ali Abdullah Saleh.
Islamic State declared its presence in November and have carried out a number of attacks since then.
Why is there fighting in Yemen?
Northern Shia Muslim rebels known as Houthis, backed by forces loyal to Yemen's ex-president, took over parts of Yemen, including the capital, Sanaa, and forced the government into exile in March
The rebels accused the government of corruption and of planning to marginalise their heartland within a proposed federal system
Forces loyal to the government and southern militias regained control of Adenin July, aided by Saudi-led coalition air strikes and troops
The UN says that fighting on the ground and air strikes in Yemen have killed more than 5,700 people since March