A day after the two bombs, which appeared to be a co-ordinated attack on police, an offshoot of the Kurdistan Workers Party (PKK) said it was responsible.
In a statement on its website, the group, called Kurdistan Freedom Hawks, or TAK, also said two of its militants were killed.
As some of the victims were laid to rest, Turkey vowed to make the perpetrators pay.
“Sooner or later, we will have our vengeance,” said interior minister Suleyman Soylu, speaking at a funeral service at Istanbul police headquarters.
“This blood will not be left on the ground, no matter what the price, what the cost.”
President Tayyip Erdogan called the attack on Saturday night “the ugly face of terror” and vowed to fight “the curse of terrorism till the end”.
In the first explosion, a car bomb was blown up outside the Vodafone Arena, home to the club Besiktas, after a football match. Less than a minute later, a suicide bomber hit in an adjacent park.
Thirteen people have been arrested as officials started pointing to Kurdish militants soon after the attack.
Most of the victims – 30 – were police officers, seven were civilians and one was yet to be identified, officials said.
More than 150 were injured, and 136 people remained in hospital, including 14 under intensive care.
Authorities have determined that about 300-400kg of explosives were used in the powerful blasts.
The number of civilians killed was lower than feared as fans had already left the stadium after the game when the blasts occurred.
The first explosion struck in an area where police special forces were located at the stadium exit. A riot police bus appears to have been the target, said Mr Soylu.
The suicide bomber triggered the explosive after being stopped by police in the nearby Macka Park.
Witnesses have described scenes of panic as the explosions rocked the area and shattered windows.
“It was like hell. The flames went all the way up to the sky,” said Omer Yilmaz, who works as a cleaner at the nearby Dolmabahce Mosque, directly across the road from the stadium.
“People ducked under the tables, women began crying. Football fans drinking tea at the cafe sought shelter, it was horrible.”
Sunday was a day of national mourning, with flags flying at half-mast across the nation and officials calling for a march against terrorism.
Mr Erdogan, who cancelled a planned trip to Kazakhstan, described the blasts as a terrorist attack on police and civilians. He said the aim of the bombings, two hours after the end of a match attended by thousands of people, had been to cause maximum casualties.
“We have once again witnessed tonight in Istanbul the ugly face of terror, which tramples on every value and decency,” he said.
“Nobody should doubt that with God’s will, we as a country and a nation will overcome terror, terrorist organisations.”
This year Istanbul has witnessed a spate of attacks attributed by authorities to the Islamic State group or claimed by Kurdish militants.
A state of emergency is in force following a failed coup attempt on 15 July.