Police said they had ruled out the possibility that it was an act of terrorism, and that the man appeared to be suffering from mental distress.
Simone Zucca, a waitress at the Pronto restaurant on the main square, said that she heard screaming before she saw the driver, saying: "He raced through the front of the restaurant without braking. He then caught a boy who was thrown against the wall."
Eyewitnesses recounted seeing bodies lying on the ground covering in blood, with one describing the sound of the SUV knocking chairs in the square flying as being 'like gunfire'.
The 26-year-old, who is married with two children and reportedly facing a divorce, had been evicted from his home because of domestic violence on May 28, police confirmed, saying they believe the man was struggling to cope with "family problems".
Josef Klamminger, National Police Director, said they could "absolutely rule out an extremist, religious or political background" to the attack, saying it was the act of a "lone wolf".
A surgeon who arrived on the scene shortly after the attack told the Austrian daily newspaper Kleine Zeitung he tried to revive the female cyclist "but help came too late for her".
Siegfried Nagl, mayor of Graz, witnessed the attack in his rear-view mirror while riding his Vespa motor scooter metres away from the speeding car, and only managed to escape by driving on the sidewalk.
"The driver deliberately drove into passers-by. I saw myself how a woman fell down," he said.
"This offender, the murderer first mowed down a couple of people, the man was apparently killed instantly, then I thought, he has stopped, but then he targeted me and another passer-by."
Mr Nagl said: "I've never experienced anything like it. Everything will be cancelled, all the celebrations and festivals, we will hang black flags."
More than 60 ambulances converged on the scene and the Austrian auto club sent four helicopters to bring the victims to hospitals.
The provincial governor Hermann Schützenhöfer denounced the attack as a "heinous act" for which such there is "neither an explanation nor an excuse".
"We have much to do to ensure cohesion in our community, which has clearly become difficult for many people," he said.
Austrian chancellor Werner Faymann said he was "shocked" by the attack, and said his sympathy and thoughts were with the victims and their families in this difficult time.
A memorial for the victims of the carnage is expected to take place on Sunday evening.