The blast hit the Karrada neighbourhood in the early hours of the day, as the area was packed with shoppers after sundown ahead of this week's holiday marking the end of the Muslim fasting month of Ramazan.
It came a week after Iraqi security forces recaptured the city of Fallujah from IS, leaving Mosul as the only Iraqi city under the jihadists' control.
The bombing also wounded more than 130 people, security and medical officials said.
Prime Minister Haider al-Abadi visited the site of the bombing and vowed "punishment" for its perpetrators, his office said.
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The attack set buildings in the area ablaze, reducing some to charred hulks and also torching shops.
Men carried the bodies of two victims out of one burned building and a crowd of people looked on from the rubble-filled street as firefighters worked at the site.
IS issued a statement claiming responsiblity for the suicide car bombing, saying it was carried out by an Iraqi as part of the group's "ongoing security operations".
The group said the bombing targeted members of Iraq's Shia Muslim majority, whom the Sunni extremists consider heretics and frequently attack in Baghdad and elsewhere.
Bombings in the capital have decreased since IS overran large areas north and west of Baghdad in June 2014, apparently because the jihadists were occupied with operations elsewhere.
But the group has struck back against Iraqi civilians after suffering military setbacks, both as revenge and to portray itself as being on the offensive.
In May, the capital was rocked by a series of blasts that killed more than 150 people in seven days.
Hostage-takers were from Bangladesh group, not Islamic State
With thousands of vehicles moving in and out of the city each day, such bombings are difficult to prevent.
But there are also flaws in Iraqi security measures in the city, especially the continued use of fake bomb detectors at checkpoints in the city years after the man who sold them to Iraq was jailed for fraud in Britain.
A video posted on social media Sunday showed men - apparently angry at the government's failure to prevent the attack - throwing rocks toward what was said to be Abadi's convoy.
Iraqi forces completely recaptured Fallujah, a city 50 kilometres (30 miles) west of Baghdad, from the jihadists a week ago.
Anti-government fighters seized Fallujah in early 2014 and it later became one of IS's main strongholds in the country.
Iraqi forces launched an operation in May to recapture the city, one of only two in the country then held by IS.
The defeat was compounded by a devastating series of air strikes targeting jihadist forces as they sought to flee the Fallujah area.
Iraqi and US-led coalition aircraft destroyed hundreds of IS vehicles and killed dozens of fighters in two days of strikes against convoys after the end of the fighting, officials said.
With Fallujah retaken, Iraqi forces are now setting their sights on second city Mosul, the last major population centre held by IS in Iraq.
Initial operations aimed at setting the stage for a final assault on the city have begun, and the US-led coalition is carrying out strikes in the area.
The Pentagon announced on Friday that the coalition had killed two senior IS leaders in the Mosul area the week before.
In addition to Mosul, IS still holds significant territory in Nineveh province, of which it is the capital, as well as areas in Kirkuk to its west and Anbar to its south.
The jihadist group seized control of large parts of Iraq and Syria in mid-2014, declaring an Islamic "caliphate", committing widespread atrocities and organising or inspiring a series of deadly attacks in Western cities and across the Middle East.
Pakistan condemns attack
Pakistan has strongly condemned the terrorist attack.
“The government and the people of Pakistan extend their heartfelt condolences to the brotherly government and people of Iraq over the loss of innocent lives,” said a statement issued by the Foreign Office. “Our hearts go out to the bereaved families and we wish speedy recovery for the injured.”
Pakistan stands in solidarity with the brotherly people of Iraq in this hour of grief, the statement added.
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