Local media reported that the blast was remotely detonated as the bus, carrying a group of riot police, passed. No group has taken responsibility for the bombing.
The area surrounding the explosion -a busy neighborhood that includes the city's landmark Beyazit Square, the main Istanbul University campus and the Vezneciler metro station - was evacuated following the blast.
Images from the site of the explosion showed visible damage to surrounding buildings, with windows blown out and glass strewn across the street.
The United States, France and Germany were quick to condemn the attack, with French President Francois Hollande saying in a statement that "this intolerable act of violence must more than ever confirm our mutual determination to fight all forms of terrorism."
The U.S. Embassy in Turkey said on Twitter that it would "continue to stand shoulder-to-shoulder with Turkey in the fight against terrorism."
Turkey has been rocked by a string of terror attacks over the past year as it weathers bombing campaigns carried out both by ISIS and Kurdish militants.
The violence has had an impact on Turkey's tourism industry, a key sector of the national economy.