The explosions, believed to be a car bomb and a suicide bomb, targeted police officers, officials say.
Witnesses heard gunfire after the attack, which came two hours after fans had left the Besiktas stadium.
Ten arrests were made. Turkey has seen a recent spate of deadly militant attacks in major cities.
No group has said it was behind the attack but a wave of bombings in Turkey this year has been carried out by Kurdish militants and so-called Islamic State (IS), the BBC's Turkey correspondent, Mark Lowen, reports.
President Recep Tayyip Erdogan confirmed fatalities but gave no details.
"A terrorist attack has been carried out against our security forces and our citizens," he said.
"It has been understood that the explosions after the Besiktas-Bursaspor football game aimed to maximise casualties. As a result of these attacks unfortunately we have martyrs and wounded."
The blasts came two hours after a football match at the Vodafone Arena between Besiktas and Bursaspor, two of Turkey's top teams, ended.
Local media reported that fans had already dispersed. Bursaspor posted on Twitter that none of its groups knew of any injured fans.
Interior Minister Suleyman Soylu said: "It is thought to be a car bomb at a point where our special forces police were located, right after the match at the exit where Bursaspor fans exited, after the fans had left."
He later said the second blast, at nearby Macka Park, was believed to be a suicide bomb.
TV channel NTV reported that the target of the first attack was a bus carrying riot police.
Photographs posted on Instagram after the explosion showed helmets strewn on a road and damaged vehicles.
A police helicopter circled overhead and windows in nearby buildings were blown out by the force of the blasts.