The suspect, a 27-year-old man named Chang Wen, fell to his death from a department store building in Taipei after the attack on Friday.
The assaults shocked Taiwan, where violent crime is rare, prompting authorities to step up security at crowded spots and big events, including its New Year's Eve celebration that usually attracts many locals and tourists to an outdoor countdown.
Chang Jung-Hsin, director-general of the National Police Agency, said the suspect began a series of attacks in the midafternoon, first setting fire to roads and causing damage to cars and motorbikes.
He also set fire to where he lived.
The suspect later threw smoke grenades near two exits of the Taipei Main metro station and used a knife to fatally wound a person, Chang said.
After the attack, the suspect took an underground path to return to the hotel where he was staying.
Then he threw more smoke grenades and used his knife to kill another person outside the Eslite Spectrum Nanxi department store, the official said.
He also fatally wounded another person on the fourth floor of the department store building before he jumped from the fifth floor to his death, Chang said.
The official said the smoke grenades the assailant used could be purchased online and that the suspect visited the sites before the attack.
"The suspect planned an indiscriminate attack. He acted according to his plan," Chang said, adding the initial investigation did not find that the suspect acted with anyone else.
Authorities were still investigating his motive.
The suspect had voluntarily served in the military but was fired for drunk driving, Chang said.
The suspect was wanted from July after failing to report to military service, he said.
"The suspect hasn't contacted his family for more than two years," Chang said.
"They said the suspect has been interested in guns and weapons since he was young."
Six injured people were still in the hospital.
Two were being treated in intensive care units, but their conditions were stable after surgeries, authorities said.
After receiving briefings from officials, Taiwanese President Lai Ching-te called for a thorough investigation into the incident, looking into the suspect's background, motives, financial resources and whether he received help from others.
