Officers took a suspect into custody after shooting him in the leg, police said. Friday night, state broadcaster YLE reported police had taken several other suspects into custody.
Police have not revealed a motive and said it was too early to tell whether the attack was terror-related.
"We have a heightened awareness level of terrorism currently, but we did not have any specific threats to act upon," Turku Superintendent Stephan Sundqvist told CNN.
The attack prompted police to boost patrols across the country, and security has been tightened at the Helsinki-Vantaa airport and train stations. Police said on Twitter that they were also patrolling a shopping mall.
Finnish Interior Minister Paula Risikko told YLE the identity of the attacker was still unknown.
'There was blood everywhere'
The attack occurred around 4 p.m. local time (9 a.m. ET) at two marketplaces close to each other in the city center -- Kauppatori, known in English as Central Market Square, as well as the Puutori Market Square, police said.
Kent Svensson, 44, of Sweden said he witnessed a man with "a huge white knife" running and stabbing people in his path.
"It was really horrible. We were sitting on a terrace just next to the square and this woman just screamed like hell and this guy was standing in front of her with a huge knife just stabbing people," he said.
"There was blood everywhere," he said, adding that he saw a victim lying on the ground.
Finnish media published photographs of the attack site, and one showed a body covered with a white sheet. Others showed a small crowd around someone apparently bleeding on a cobblestone road and several ambulances around a cordoned-off area.
Three people are now undergoing surgery, YLE reported, citing hospital sources.
Police warned the public to stay away from the city center.
Mayor urges calm
Turku sits on the coast around 140 kilometers (around 85 miles) west of the capital, Helsinki. With a population of about 185,000, it's an important business and cultural center.
Sundqvist said police were on the scene within a few minutes of the first reports of the stabbings.
"There was only one shot fired and the aggressor was apprehended and brought to hospital," he said.
The mayor of Turku, Aleksi Randell, said the city center appears to be safe now and urged citizens to remain calm.
"I want to say this to all our residents: It is difficult to understand that such violence would happen on this scale in Turku," he said in a statement posted on the city website.
Occurrences that have been all too frequent in Europe and around the world have now arrived here, which is what (we've) been afraid of."
The Finnish government convened Friday to discuss the attack and condemned it.
"The act is utterly deplorable and an exceptionally serious violent offense in the Finnish context -- it is deeply shocking for us all," Prime Minister Juha Sipilä said on the government website.