A statement posted on Twitter by the jihadist group said it was behind a blast that hit the convoy of Jaafar Saad in the Tawahi neighbourhood of the major port city, killing him and eight of his bodyguards.
In a statement carried by the official Saba news agency, the head of security in Aden General Mohamed Mussad confirmed Saad's death and said six of his guards were also killed.
Images circulated on social media after the attack showed a wrecked car on fire on a main road in the city.
Saad was only recently appointed governor and was known to be close to President Abedrabbo Mansour Hadi, who returned to Aden last month after several months in exile in Riyadh.
Pro-Hadi forces, aided by a Saudi-led coalition, have battled Iran-backed rebels in Yemen since March, after the insurgents overran the capital Sanaa and advanced south, forcing the government to flee to Saudi Arabia.
Fighting between the Huthi rebels and loyalist forces has plunged the impoverished nation into chaos, which jihadist groups have exploited to make sweeping gains, particularly in southern regions.
Tawahi itself has become a known hideout for jihadists, including militants belonging to Al-Qaeda.
IS has claimed a string of attacks in Yemen, including the bombing of Hadi's government headquarters in October and multiple suicide attacks on mosques in Sanaa attended by Shiite worshippers that killed scores.
Saad's death comes a day after Yemen's UN envoy held talks with Hadi in Aden aimed at kickstarting peace talks between the warring sides.
Ismail Ould Cheikh Ahmed met Hadi to seek his agreement to convene negotiations with the rebels in Geneva next week, an official close to the president told AFP.
But the mission was "difficult", said the source, accusing the rebels of dragging their feet on implementing UN Security Council Resolution 2216, which calls for them to withdraw from occupied territory.
And Foreign Minister Abdel Malak al-Mekhlafi told AFP: "The putschists are refusing to lay down their arms or to allow the government to carry out its duties" from Sanaa.
"They have not announced their list of negotiators" for the talks "and are trying to escalate the situation on the ground by bombing residential districts of Taez", a strategic city in southwest Yemen under siege by the rebels and their allies, he said.
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