Police said they had identified the suspect, who was shot dead by armed officers in the early hours of Sunday, but they have not released his name.
The head of Denmark’s Security and Intelligence Service, Jens Madsen, said the gunman was known to his agency before the shooting, and probably acted alone.
Mr Madsen said: “We cannot yet say anything concrete about the motive … but are considering that he might have been inspired by the events in Paris some weeks ago.”
Danish authorities have been on alert since Islamist gunmen killed 17 people in three days of violence in Paris in January that began with an attack on the satirical magazine Charlie Hebdo.
Police who earlier released a photo of the suspect dressed in a heavy winter coat and mask said they did not believe he had received training in jihadist camps in the Middle East.
The gunman fired up to 40 shots at the Krudttonden cafe during a free speech event attended by the Swedish artist Lars Vilks, who has received death threats for publishing cartoons of the prophet Mohammed.
Vilks was unharmed but a 55-year-old film director, Finn Norgaard, died and three police officers were injured.
Hours later a man guarding a Jewish community centre during a bar mitzvah party was shot in the head. He has been identified as 37-year-old Dan Uzan.
Rabbi Jair Melchior, Denmark’s chief rabbi, said Mr Uzan was an “irreplaceable” security guard protecting the city’s Jewish community, adding: “He was a person who was always willing to help – an amazing, amazing guy.”
Prime Minister Helle Thorning-Schmidt said the attacks were terrorism and promised to protect freedom of speech and Denmark’s Jewish community.
She said: “When you mercilessly fire deadly bullets at innocent people taking part in a debate, when you attack the Jewish community, you attack our democracy.
“We will do everything possible to protect our Jewish community.”