Mohamed Geele, 29, may face a 12-year sentence in prison for his attack on Kurt Westergaard, 75, who avoided injury by sheltering in a panic room.
But he was acquitted of a separate charge of attempting to murder a policeman trying to arrest him.
Geele, who was wounded by police after the attack, denied the charges.
He told the court in Aarhus he was only trying to frighten the cartoonist.
A judge is expected to pass sentence on Friday.
Mr Westergaard’s cartoon of a turban bomb was one of 12 published in Danish newspaper Jyllands-Posten in 2005 that prompted protests among Muslims around the world.
He first went into hiding but then decided to live openly in a heavily fortified house in Aarhus, Denmark’s second city.
Axe and knife
Judge Ingrid Thorsboe told the court that Geele’s actions “must be considered as an attempt to instil a heightened level of fear in the population and to destabilise the structures of society”.
The maximum penalty Geele faced was a life sentence but prosecutors have asked for 12 years, followed by deportation to Somalia.
His defence lawyer, Niels Strauss, asked for, at most, a suspended six-year sentence and for his client not to be deported.
During the trial, the court heard that Geele broke into Mr Westergaard’s house in Aarhus on New Year’s Day 2010, smashing the door down.
Mr Westergaard was at home with his five-year-old grand-daughter Stephanie at the time.
Geele, who was armed with an axe and a knife, screamed “You must die!” and “You are going to Hell!”, the cartoonist told the court.
He locked himself in his panic room – a reinforced bathroom – leaving Stephanie alone in the living-room because his attacker was “after me, not the people around me”.
The little girl, who was unhurt in the attack, testified that she had thought Geele was a thief and had asked him to go away.
Geele tried to get into the bathroom, hacking at the door with his axe, but fled when he heard police sirens.
Still armed, he was shot and wounded by police arriving at the scene, who then arrested him.
Geele had searched for the cartoonist’s address on the internet and, on New Year’s Eve, bought the axe and sharpened the knife, which he already possessed.
While acquitted of the charge of attempting to murder a policeman, brought because he threw his axe at one of the officers who arrived to arrest him, Geele was found guilty of aggravated assault on the officer.
Source: BBC